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Properties

Label 2-5e2-25.22-c2-0-2
Degree 22
Conductor 2525
Sign 0.981+0.193i0.981 + 0.193i
Analytic cond. 0.6812000.681200
Root an. cond. 0.8253480.825348
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.972 − 0.495i)2-s + (0.872 − 0.138i)3-s + (−1.65 + 2.27i)4-s + (−2.66 − 4.23i)5-s + (0.779 − 0.566i)6-s + (1.62 + 1.62i)7-s + (−1.16 + 7.34i)8-s + (−7.81 + 2.54i)9-s + (−4.68 − 2.79i)10-s + (3.53 − 10.8i)11-s + (−1.12 + 2.20i)12-s + (7.63 + 3.89i)13-s + (2.39 + 0.776i)14-s + (−2.90 − 3.32i)15-s + (−0.963 − 2.96i)16-s + (25.1 + 3.98i)17-s + ⋯
L(s)  = 1  + (0.486 − 0.247i)2-s + (0.290 − 0.0460i)3-s + (−0.412 + 0.567i)4-s + (−0.532 − 0.846i)5-s + (0.129 − 0.0944i)6-s + (0.232 + 0.232i)7-s + (−0.145 + 0.917i)8-s + (−0.868 + 0.282i)9-s + (−0.468 − 0.279i)10-s + (0.320 − 0.987i)11-s + (−0.0938 + 0.184i)12-s + (0.587 + 0.299i)13-s + (0.170 + 0.0554i)14-s + (−0.193 − 0.221i)15-s + (−0.0602 − 0.185i)16-s + (1.47 + 0.234i)17-s + ⋯

Functional equation

Λ(s)=(25s/2 ΓC(s) L(s)=((0.981+0.193i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(25s/2 ΓC(s+1) L(s)=((0.981+0.193i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2525    =    525^{2}
Sign: 0.981+0.193i0.981 + 0.193i
Analytic conductor: 0.6812000.681200
Root analytic conductor: 0.8253480.825348
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ25(22,⋅)\chi_{25} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 25, ( :1), 0.981+0.193i)(2,\ 25,\ (\ :1),\ 0.981 + 0.193i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.06696−0.104475i1.06696 - 0.104475i
L(12)L(\frac12) ≈\approx 1.06696−0.104475i1.06696 - 0.104475i
L(2)L(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+(2.66+4.23i)T 1 + (2.66 + 4.23i)T
good2 1+(−0.972+0.495i)T+(2.35−3.23i)T2 1 + (-0.972 + 0.495i)T + (2.35 - 3.23i)T^{2}
3 1+(−0.872+0.138i)T+(8.55−2.78i)T2 1 + (-0.872 + 0.138i)T + (8.55 - 2.78i)T^{2}
7 1+(−1.62−1.62i)T+49iT2 1 + (-1.62 - 1.62i)T + 49iT^{2}
11 1+(−3.53+10.8i)T+(−97.8−71.1i)T2 1 + (-3.53 + 10.8i)T + (-97.8 - 71.1i)T^{2}
13 1+(−7.63−3.89i)T+(99.3+136.i)T2 1 + (-7.63 - 3.89i)T + (99.3 + 136. i)T^{2}
17 1+(−25.1−3.98i)T+(274.+89.3i)T2 1 + (-25.1 - 3.98i)T + (274. + 89.3i)T^{2}
19 1+(5.60+7.71i)T+(−111.+343.i)T2 1 + (5.60 + 7.71i)T + (-111. + 343. i)T^{2}
23 1+(5.39+10.5i)T+(−310.+427.i)T2 1 + (5.39 + 10.5i)T + (-310. + 427. i)T^{2}
29 1+(−5.56+7.65i)T+(−259.−799.i)T2 1 + (-5.56 + 7.65i)T + (-259. - 799. i)T^{2}
31 1+(42.2−30.7i)T+(296.−913.i)T2 1 + (42.2 - 30.7i)T + (296. - 913. i)T^{2}
37 1+(21.6−42.4i)T+(−804.−1.10e3i)T2 1 + (21.6 - 42.4i)T + (-804. - 1.10e3i)T^{2}
41 1+(−16.6−51.2i)T+(−1.35e3+988.i)T2 1 + (-16.6 - 51.2i)T + (-1.35e3 + 988. i)T^{2}
43 1+(−46.5+46.5i)T−1.84e3iT2 1 + (-46.5 + 46.5i)T - 1.84e3iT^{2}
47 1+(8.92+56.3i)T+(−2.10e3+682.i)T2 1 + (8.92 + 56.3i)T + (-2.10e3 + 682. i)T^{2}
53 1+(17.7−2.81i)T+(2.67e3−868.i)T2 1 + (17.7 - 2.81i)T + (2.67e3 - 868. i)T^{2}
59 1+(13.2−4.30i)T+(2.81e3−2.04e3i)T2 1 + (13.2 - 4.30i)T + (2.81e3 - 2.04e3i)T^{2}
61 1+(−0.671+2.06i)T+(−3.01e3−2.18e3i)T2 1 + (-0.671 + 2.06i)T + (-3.01e3 - 2.18e3i)T^{2}
67 1+(−116.−18.4i)T+(4.26e3+1.38e3i)T2 1 + (-116. - 18.4i)T + (4.26e3 + 1.38e3i)T^{2}
71 1+(57.1+41.5i)T+(1.55e3+4.79e3i)T2 1 + (57.1 + 41.5i)T + (1.55e3 + 4.79e3i)T^{2}
73 1+(−28.5−56.0i)T+(−3.13e3+4.31e3i)T2 1 + (-28.5 - 56.0i)T + (-3.13e3 + 4.31e3i)T^{2}
79 1+(−19.9+27.4i)T+(−1.92e3−5.93e3i)T2 1 + (-19.9 + 27.4i)T + (-1.92e3 - 5.93e3i)T^{2}
83 1+(7.27−45.9i)T+(−6.55e3−2.12e3i)T2 1 + (7.27 - 45.9i)T + (-6.55e3 - 2.12e3i)T^{2}
89 1+(30.3+9.84i)T+(6.40e3+4.65e3i)T2 1 + (30.3 + 9.84i)T + (6.40e3 + 4.65e3i)T^{2}
97 1+(−10.7−67.9i)T+(−8.94e3+2.90e3i)T2 1 + (-10.7 - 67.9i)T + (-8.94e3 + 2.90e3i)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

−17.09031050092095685258334964200, −16.33608268298649404076872121318, −14.52941563581730054658962644616, −13.55077643561507349917876369398, −12.28373900826838493943046107872, −11.31096096045243929638036204193, −8.824028590629797724861334441435, −8.162261812495775806862796431063, −5.36767013064634774912873572999, −3.54955360399321777443784806305, 3.78551683894993114844835411628, 5.85932041774539738289013968267, 7.59625204027150619256792919361, 9.480997137802061516929645504705, 10.89094010768245319584072984656, 12.45954206701400894343342456306, 14.27257112577544554237107433762, 14.55601403703903303004904591631, 15.79404460772429460552614651469, 17.56637785696073194779884322767

Graph of the ZZ-function along the critical line