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Properties

Label 2-115-115.107-c1-0-8
Degree 22
Conductor 115115
Sign 0.746+0.665i0.746 + 0.665i
Analytic cond. 0.9182790.918279
Root an. cond. 0.9582690.958269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.63 − 1.22i)2-s + (1.28 + 0.480i)3-s + (0.607 − 2.07i)4-s + (−2.17 + 0.527i)5-s + (2.69 − 0.790i)6-s + (−2.19 + 0.476i)7-s + (−0.112 − 0.301i)8-s + (−0.835 − 0.723i)9-s + (−2.90 + 3.51i)10-s + (0.377 − 0.0542i)11-s + (1.77 − 2.37i)12-s + (0.0183 − 0.0842i)13-s + (−2.99 + 3.45i)14-s + (−3.05 − 0.365i)15-s + (3.07 + 1.97i)16-s + (1.64 − 3.00i)17-s + ⋯
L(s)  = 1  + (1.15 − 0.864i)2-s + (0.744 + 0.277i)3-s + (0.303 − 1.03i)4-s + (−0.971 + 0.235i)5-s + (1.09 − 0.322i)6-s + (−0.828 + 0.180i)7-s + (−0.0397 − 0.106i)8-s + (−0.278 − 0.241i)9-s + (−0.918 + 1.11i)10-s + (0.113 − 0.0163i)11-s + (0.513 − 0.686i)12-s + (0.00508 − 0.0233i)13-s + (−0.800 + 0.923i)14-s + (−0.789 − 0.0943i)15-s + (0.769 + 0.494i)16-s + (0.398 − 0.728i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((0.746+0.665i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+1/2) L(s)=((0.746+0.665i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 0.746+0.665i0.746 + 0.665i
Analytic conductor: 0.9182790.918279
Root analytic conductor: 0.9582690.958269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ115(107,⋅)\chi_{115} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :1/2), 0.746+0.665i)(2,\ 115,\ (\ :1/2),\ 0.746 + 0.665i)

Particular Values

L(1)L(1) ≈\approx 1.67857−0.639741i1.67857 - 0.639741i
L(12)L(\frac12) ≈\approx 1.67857−0.639741i1.67857 - 0.639741i
L(32)L(\frac{3}{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+(2.17−0.527i)T 1 + (2.17 - 0.527i)T
23 1+(−4.78−0.374i)T 1 + (-4.78 - 0.374i)T
good2 1+(−1.63+1.22i)T+(0.563−1.91i)T2 1 + (-1.63 + 1.22i)T + (0.563 - 1.91i)T^{2}
3 1+(−1.28−0.480i)T+(2.26+1.96i)T2 1 + (-1.28 - 0.480i)T + (2.26 + 1.96i)T^{2}
7 1+(2.19−0.476i)T+(6.36−2.90i)T2 1 + (2.19 - 0.476i)T + (6.36 - 2.90i)T^{2}
11 1+(−0.377+0.0542i)T+(10.5−3.09i)T2 1 + (-0.377 + 0.0542i)T + (10.5 - 3.09i)T^{2}
13 1+(−0.0183+0.0842i)T+(−11.8−5.40i)T2 1 + (-0.0183 + 0.0842i)T + (-11.8 - 5.40i)T^{2}
17 1+(−1.64+3.00i)T+(−9.19−14.3i)T2 1 + (-1.64 + 3.00i)T + (-9.19 - 14.3i)T^{2}
19 1+(2.64+0.777i)T+(15.9+10.2i)T2 1 + (2.64 + 0.777i)T + (15.9 + 10.2i)T^{2}
29 1+(−0.489−1.66i)T+(−24.3+15.6i)T2 1 + (-0.489 - 1.66i)T + (-24.3 + 15.6i)T^{2}
31 1+(1.34−2.94i)T+(−20.3−23.4i)T2 1 + (1.34 - 2.94i)T + (-20.3 - 23.4i)T^{2}
37 1+(−0.849−11.8i)T+(−36.6+5.26i)T2 1 + (-0.849 - 11.8i)T + (-36.6 + 5.26i)T^{2}
41 1+(5.72+6.61i)T+(−5.83+40.5i)T2 1 + (5.72 + 6.61i)T + (-5.83 + 40.5i)T^{2}
43 1+(0.439−1.17i)T+(−32.4−28.1i)T2 1 + (0.439 - 1.17i)T + (-32.4 - 28.1i)T^{2}
47 1+(−7.76−7.76i)T+47iT2 1 + (-7.76 - 7.76i)T + 47iT^{2}
53 1+(2.77+12.7i)T+(−48.2+22.0i)T2 1 + (2.77 + 12.7i)T + (-48.2 + 22.0i)T^{2}
59 1+(6.76+10.5i)T+(−24.5+53.6i)T2 1 + (6.76 + 10.5i)T + (-24.5 + 53.6i)T^{2}
61 1+(4.28+1.95i)T+(39.9+46.1i)T2 1 + (4.28 + 1.95i)T + (39.9 + 46.1i)T^{2}
67 1+(−0.294−0.392i)T+(−18.8+64.2i)T2 1 + (-0.294 - 0.392i)T + (-18.8 + 64.2i)T^{2}
71 1+(−0.242+1.68i)T+(−68.1−20.0i)T2 1 + (-0.242 + 1.68i)T + (-68.1 - 20.0i)T^{2}
73 1+(8.94−4.88i)T+(39.4−61.4i)T2 1 + (8.94 - 4.88i)T + (39.4 - 61.4i)T^{2}
79 1+(−4.54+2.92i)T+(32.8−71.8i)T2 1 + (-4.54 + 2.92i)T + (32.8 - 71.8i)T^{2}
83 1+(−15.3+1.09i)T+(82.1−11.8i)T2 1 + (-15.3 + 1.09i)T + (82.1 - 11.8i)T^{2}
89 1+(−1.74−3.81i)T+(−58.2+67.2i)T2 1 + (-1.74 - 3.81i)T + (-58.2 + 67.2i)T^{2}
97 1+(1.07+0.0772i)T+(96.0+13.8i)T2 1 + (1.07 + 0.0772i)T + (96.0 + 13.8i)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

−13.39052058358060883059106338413, −12.40640831231717928952678948584, −11.66488241137958127126092037056, −10.63096725835839958092670850641, −9.305686574891183092914120807652, −8.149456694493357120116280013299, −6.58191869149000479975146025109, −4.88101420264745926110859556909, −3.52777648782859861639968284789, −2.92133048819605786260221828754, 3.21626170052963107240049121473, 4.26218028870937390123560525558, 5.77425415506106517338138041745, 7.05846838529320989226878296400, 7.905802676583933518079823603253, 9.058621318574708772931916829562, 10.70057164660134706409397422993, 12.20161304985985350695706705056, 12.96654405171295354384953708883, 13.71141741762519387255016078970

Graph of the ZZ-function along the critical line