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Properties

Label 2-115-115.53-c3-0-17
Degree 22
Conductor 115115
Sign 0.978+0.206i0.978 + 0.206i
Analytic cond. 6.785216.78521
Root an. cond. 2.604842.60484
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.00959 + 0.134i)2-s + (−1.51 − 0.329i)3-s + (7.90 − 1.13i)4-s + (11.1 − 0.644i)5-s + (0.0297 − 0.206i)6-s + (3.12 + 1.70i)7-s + (0.457 + 2.10i)8-s + (−22.3 − 10.2i)9-s + (0.193 + 1.49i)10-s + (12.1 − 10.5i)11-s + (−12.3 − 0.883i)12-s + (20.9 + 38.4i)13-s + (−0.198 + 0.435i)14-s + (−17.1 − 2.70i)15-s + (60.9 − 17.9i)16-s + (79.5 − 59.5i)17-s + ⋯
L(s)  = 1  + (0.00339 + 0.0474i)2-s + (−0.291 − 0.0634i)3-s + (0.987 − 0.141i)4-s + (0.998 − 0.0576i)5-s + (0.00202 − 0.0140i)6-s + (0.168 + 0.0920i)7-s + (0.0201 + 0.0928i)8-s + (−0.828 − 0.378i)9-s + (0.00612 + 0.0471i)10-s + (0.334 − 0.289i)11-s + (−0.297 − 0.0212i)12-s + (0.447 + 0.820i)13-s + (−0.00379 + 0.00831i)14-s + (−0.294 − 0.0465i)15-s + (0.952 − 0.279i)16-s + (1.13 − 0.849i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((0.978+0.206i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+3/2) L(s)=((0.978+0.206i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 0.978+0.206i0.978 + 0.206i
Analytic conductor: 6.785216.78521
Root analytic conductor: 2.604842.60484
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ115(53,⋅)\chi_{115} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :3/2), 0.978+0.206i)(2,\ 115,\ (\ :3/2),\ 0.978 + 0.206i)

Particular Values

L(2)L(2) ≈\approx 2.09169−0.218262i2.09169 - 0.218262i
L(12)L(\frac12) ≈\approx 2.09169−0.218262i2.09169 - 0.218262i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(−11.1+0.644i)T 1 + (-11.1 + 0.644i)T
23 1+(−48.5−99.0i)T 1 + (-48.5 - 99.0i)T
good2 1+(−0.00959−0.134i)T+(−7.91+1.13i)T2 1 + (-0.00959 - 0.134i)T + (-7.91 + 1.13i)T^{2}
3 1+(1.51+0.329i)T+(24.5+11.2i)T2 1 + (1.51 + 0.329i)T + (24.5 + 11.2i)T^{2}
7 1+(−3.12−1.70i)T+(185.+288.i)T2 1 + (-3.12 - 1.70i)T + (185. + 288. i)T^{2}
11 1+(−12.1+10.5i)T+(189.−1.31e3i)T2 1 + (-12.1 + 10.5i)T + (189. - 1.31e3i)T^{2}
13 1+(−20.9−38.4i)T+(−1.18e3+1.84e3i)T2 1 + (-20.9 - 38.4i)T + (-1.18e3 + 1.84e3i)T^{2}
17 1+(−79.5+59.5i)T+(1.38e3−4.71e3i)T2 1 + (-79.5 + 59.5i)T + (1.38e3 - 4.71e3i)T^{2}
19 1+(3.75+26.0i)T+(−6.58e3+1.93e3i)T2 1 + (3.75 + 26.0i)T + (-6.58e3 + 1.93e3i)T^{2}
29 1+(141.+20.3i)T+(2.34e4+6.87e3i)T2 1 + (141. + 20.3i)T + (2.34e4 + 6.87e3i)T^{2}
31 1+(183.+118.i)T+(1.23e4+2.70e4i)T2 1 + (183. + 118. i)T + (1.23e4 + 2.70e4i)T^{2}
37 1+(150.−56.1i)T+(3.82e4−3.31e4i)T2 1 + (150. - 56.1i)T + (3.82e4 - 3.31e4i)T^{2}
41 1+(−112.−245.i)T+(−4.51e4+5.20e4i)T2 1 + (-112. - 245. i)T + (-4.51e4 + 5.20e4i)T^{2}
43 1+(5.35−24.6i)T+(−7.23e4−3.30e4i)T2 1 + (5.35 - 24.6i)T + (-7.23e4 - 3.30e4i)T^{2}
47 1+(321.−321.i)T−1.03e5iT2 1 + (321. - 321. i)T - 1.03e5iT^{2}
53 1+(−204.+375.i)T+(−8.04e4−1.25e5i)T2 1 + (-204. + 375. i)T + (-8.04e4 - 1.25e5i)T^{2}
59 1+(−25.0+85.1i)T+(−1.72e5−1.11e5i)T2 1 + (-25.0 + 85.1i)T + (-1.72e5 - 1.11e5i)T^{2}
61 1+(277.−431.i)T+(−9.42e4−2.06e5i)T2 1 + (277. - 431. i)T + (-9.42e4 - 2.06e5i)T^{2}
67 1+(864.−61.8i)T+(2.97e5−4.28e4i)T2 1 + (864. - 61.8i)T + (2.97e5 - 4.28e4i)T^{2}
71 1+(403.−465.i)T+(−5.09e4−3.54e5i)T2 1 + (403. - 465. i)T + (-5.09e4 - 3.54e5i)T^{2}
73 1+(360.−482.i)T+(−1.09e5−3.73e5i)T2 1 + (360. - 482. i)T + (-1.09e5 - 3.73e5i)T^{2}
79 1+(1.16e3+341.i)T+(4.14e5+2.66e5i)T2 1 + (1.16e3 + 341. i)T + (4.14e5 + 2.66e5i)T^{2}
83 1+(−174.−468.i)T+(−4.32e5+3.74e5i)T2 1 + (-174. - 468. i)T + (-4.32e5 + 3.74e5i)T^{2}
89 1+(−361.+232.i)T+(2.92e5−6.41e5i)T2 1 + (-361. + 232. i)T + (2.92e5 - 6.41e5i)T^{2}
97 1+(−111.+298.i)T+(−6.89e5−5.97e5i)T2 1 + (-111. + 298. i)T + (-6.89e5 - 5.97e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.06233305926691194887323371454, −11.64555190410804812973609538132, −11.29795115499672054375946474062, −9.882684385220502005511786297863, −8.910773442253733834275503210741, −7.29845917198480675586919295253, −6.16995080414583038712995500776, −5.41896852137002351752569100777, −3.08874181581593201114293776002, −1.49032874777373313063752934026, 1.68622442522467382027371067936, 3.20817861499776172242194417619, 5.44395862306148813200386333580, 6.18732320319276988049855927878, 7.54702181743851860462796347042, 8.807039005423541932733738450021, 10.40172667873499202948394582429, 10.78238605064505427467360208624, 12.10167118343735973535481147568, 12.95325190923654764536859537162

Graph of the ZZ-function along the critical line