↑

Properties

Label 2-250-25.11-c3-0-21
Degree 22
Conductor 250250
Sign −0.926+0.377i-0.926 + 0.377i
Analytic cond. 14.750414.7504
Root an. cond. 3.840633.84063
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 − 1.17i)2-s + (3.00 − 9.24i)3-s + (1.23 − 3.80i)4-s + (−6.01 − 18.4i)6-s + 23.3·7-s + (−2.47 − 7.60i)8-s + (−54.6 − 39.7i)9-s + (6.24 − 4.53i)11-s + (−31.4 − 22.8i)12-s + (−31.6 − 23.0i)13-s + (37.8 − 27.4i)14-s + (−12.9 − 9.40i)16-s + (5.78 + 17.7i)17-s − 135.·18-s + (32.5 + 100. i)19-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.578 − 1.78i)3-s + (0.154 − 0.475i)4-s + (−0.408 − 1.25i)6-s + 1.26·7-s + (−0.109 − 0.336i)8-s + (−2.02 − 1.47i)9-s + (0.171 − 0.124i)11-s + (−0.757 − 0.550i)12-s + (−0.675 − 0.491i)13-s + (0.721 − 0.524i)14-s + (−0.202 − 0.146i)16-s + (0.0824 + 0.253i)17-s − 1.76·18-s + (0.393 + 1.20i)19-s + ⋯

Functional equation

Λ(s)=(250s/2 ΓC(s) L(s)=((−0.926+0.377i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(250s/2 ΓC(s+3/2) L(s)=((−0.926+0.377i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 250250    =    2⋅532 \cdot 5^{3}
Sign: −0.926+0.377i-0.926 + 0.377i
Analytic conductor: 14.750414.7504
Root analytic conductor: 3.840633.84063
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ250(51,⋅)\chi_{250} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 250, ( :3/2), −0.926+0.377i)(2,\ 250,\ (\ :3/2),\ -0.926 + 0.377i)

Particular Values

L(2)L(2) ≈\approx 0.604059−3.08364i0.604059 - 3.08364i
L(12)L(\frac12) ≈\approx 0.604059−3.08364i0.604059 - 3.08364i
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)
bad2 1+(−1.61+1.17i)T 1 + (-1.61 + 1.17i)T
5 1 1
good3 1+(−3.00+9.24i)T+(−21.8−15.8i)T2 1 + (-3.00 + 9.24i)T + (-21.8 - 15.8i)T^{2}
7 1−23.3T+343T2 1 - 23.3T + 343T^{2}
11 1+(−6.24+4.53i)T+(411.−1.26e3i)T2 1 + (-6.24 + 4.53i)T + (411. - 1.26e3i)T^{2}
13 1+(31.6+23.0i)T+(678.+2.08e3i)T2 1 + (31.6 + 23.0i)T + (678. + 2.08e3i)T^{2}
17 1+(−5.78−17.7i)T+(−3.97e3+2.88e3i)T2 1 + (-5.78 - 17.7i)T + (-3.97e3 + 2.88e3i)T^{2}
19 1+(−32.5−100.i)T+(−5.54e3+4.03e3i)T2 1 + (-32.5 - 100. i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1+(−127.+92.4i)T+(3.75e3−1.15e4i)T2 1 + (-127. + 92.4i)T + (3.75e3 - 1.15e4i)T^{2}
29 1+(13.9−42.9i)T+(−1.97e4−1.43e4i)T2 1 + (13.9 - 42.9i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(−42.7−131.i)T+(−2.41e4+1.75e4i)T2 1 + (-42.7 - 131. i)T + (-2.41e4 + 1.75e4i)T^{2}
37 1+(125.+91.2i)T+(1.56e4+4.81e4i)T2 1 + (125. + 91.2i)T + (1.56e4 + 4.81e4i)T^{2}
41 1+(15.6+11.3i)T+(2.12e4+6.55e4i)T2 1 + (15.6 + 11.3i)T + (2.12e4 + 6.55e4i)T^{2}
43 1−70.2T+7.95e4T2 1 - 70.2T + 7.95e4T^{2}
47 1+(139.−430.i)T+(−8.39e4−6.10e4i)T2 1 + (139. - 430. i)T + (-8.39e4 - 6.10e4i)T^{2}
53 1+(14.9−45.9i)T+(−1.20e5−8.75e4i)T2 1 + (14.9 - 45.9i)T + (-1.20e5 - 8.75e4i)T^{2}
59 1+(−635.−461.i)T+(6.34e4+1.95e5i)T2 1 + (-635. - 461. i)T + (6.34e4 + 1.95e5i)T^{2}
61 1+(−62.9+45.7i)T+(7.01e4−2.15e5i)T2 1 + (-62.9 + 45.7i)T + (7.01e4 - 2.15e5i)T^{2}
67 1+(242.+747.i)T+(−2.43e5+1.76e5i)T2 1 + (242. + 747. i)T + (-2.43e5 + 1.76e5i)T^{2}
71 1+(−289.+892.i)T+(−2.89e5−2.10e5i)T2 1 + (-289. + 892. i)T + (-2.89e5 - 2.10e5i)T^{2}
73 1+(−2.28+1.65i)T+(1.20e5−3.69e5i)T2 1 + (-2.28 + 1.65i)T + (1.20e5 - 3.69e5i)T^{2}
79 1+(−241.+744.i)T+(−3.98e5−2.89e5i)T2 1 + (-241. + 744. i)T + (-3.98e5 - 2.89e5i)T^{2}
83 1+(−169.−521.i)T+(−4.62e5+3.36e5i)T2 1 + (-169. - 521. i)T + (-4.62e5 + 3.36e5i)T^{2}
89 1+(−581.+422.i)T+(2.17e5−6.70e5i)T2 1 + (-581. + 422. i)T + (2.17e5 - 6.70e5i)T^{2}
97 1+(−175.+541.i)T+(−7.38e5−5.36e5i)T2 1 + (-175. + 541. i)T + (-7.38e5 - 5.36e5i)T^{2}
show more
show less
   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

−11.59669375882850514714565929362, −10.56256000907618187923614701737, −8.958445519002401660116192093801, −8.031965433437083236739524289218, −7.30897245473613054878910728991, −6.16751975344377210269853550022, −5.00586655068622913812021345143, −3.22589650600207751304753680562, −2.01900833222789867453948942060, −1.04236653208224314462396115976, 2.50519640889933658816930829415, 3.81258325613193288443755786566, 4.85318722684980699137940106786, 5.24622151760868161024817545830, 7.16206299535398854981017954143, 8.293416255762332703704710552589, 9.132997122429592634568326721305, 10.00117532330684402713633145320, 11.28843381035020644951223063855, 11.56706151376520245915987666943

Graph of the ZZ-function along the critical line