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Properties

Label 2-250-25.9-c1-0-0
Degree 22
Conductor 250250
Sign −0.763+0.646i-0.763 + 0.646i
Analytic cond. 1.996261.99626
Root an. cond. 1.412891.41289
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  + (−0.587 + 0.809i)2-s + (−1.63 + 0.529i)3-s + (−0.309 − 0.951i)4-s + (0.529 − 1.63i)6-s + 2.77i·7-s + (0.951 + 0.309i)8-s + (−0.0483 + 0.0351i)9-s + (−2.24 − 1.63i)11-s + (1.00 + 1.38i)12-s + (−3.33 − 4.59i)13-s + (−2.24 − 1.63i)14-s + (−0.809 + 0.587i)16-s + (−4.90 − 1.59i)17-s − 0.0597i·18-s + (−0.436 + 1.34i)19-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.941 + 0.305i)3-s + (−0.154 − 0.475i)4-s + (0.216 − 0.665i)6-s + 1.04i·7-s + (0.336 + 0.109i)8-s + (−0.0161 + 0.0117i)9-s + (−0.676 − 0.491i)11-s + (0.290 + 0.400i)12-s + (−0.925 − 1.27i)13-s + (−0.599 − 0.435i)14-s + (−0.202 + 0.146i)16-s + (−1.18 − 0.386i)17-s − 0.0140i·18-s + (−0.100 + 0.308i)19-s + ⋯

Functional equation

Λ(s)=(250s/2 ΓC(s) L(s)=((−0.763+0.646i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(250s/2 ΓC(s+1/2) L(s)=((−0.763+0.646i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 250250    =    2⋅532 \cdot 5^{3}
Sign: −0.763+0.646i-0.763 + 0.646i
Analytic conductor: 1.996261.99626
Root analytic conductor: 1.412891.41289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ250(49,⋅)\chi_{250} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 250, ( :1/2), −0.763+0.646i)(2,\ 250,\ (\ :1/2),\ -0.763 + 0.646i)

Particular Values

L(1)L(1) ≈\approx 0.0330023−0.0900521i0.0330023 - 0.0900521i
L(12)L(\frac12) ≈\approx 0.0330023−0.0900521i0.0330023 - 0.0900521i
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)
bad2 1+(0.587−0.809i)T 1 + (0.587 - 0.809i)T
5 1 1
good3 1+(1.63−0.529i)T+(2.42−1.76i)T2 1 + (1.63 - 0.529i)T + (2.42 - 1.76i)T^{2}
7 1−2.77iT−7T2 1 - 2.77iT - 7T^{2}
11 1+(2.24+1.63i)T+(3.39+10.4i)T2 1 + (2.24 + 1.63i)T + (3.39 + 10.4i)T^{2}
13 1+(3.33+4.59i)T+(−4.01+12.3i)T2 1 + (3.33 + 4.59i)T + (-4.01 + 12.3i)T^{2}
17 1+(4.90+1.59i)T+(13.7+9.99i)T2 1 + (4.90 + 1.59i)T + (13.7 + 9.99i)T^{2}
19 1+(0.436−1.34i)T+(−15.3−11.1i)T2 1 + (0.436 - 1.34i)T + (-15.3 - 11.1i)T^{2}
23 1+(0.384−0.529i)T+(−7.10−21.8i)T2 1 + (0.384 - 0.529i)T + (-7.10 - 21.8i)T^{2}
29 1+(1.26+3.89i)T+(−23.4+17.0i)T2 1 + (1.26 + 3.89i)T + (-23.4 + 17.0i)T^{2}
31 1+(2.20−6.77i)T+(−25.0−18.2i)T2 1 + (2.20 - 6.77i)T + (-25.0 - 18.2i)T^{2}
37 1+(0.615+0.847i)T+(−11.4+35.1i)T2 1 + (0.615 + 0.847i)T + (-11.4 + 35.1i)T^{2}
41 1+(7.36−5.35i)T+(12.6−38.9i)T2 1 + (7.36 - 5.35i)T + (12.6 - 38.9i)T^{2}
43 1−9.24iT−43T2 1 - 9.24iT - 43T^{2}
47 1+(2.63−0.857i)T+(38.0−27.6i)T2 1 + (2.63 - 0.857i)T + (38.0 - 27.6i)T^{2}
53 1+(−0.500+0.162i)T+(42.8−31.1i)T2 1 + (-0.500 + 0.162i)T + (42.8 - 31.1i)T^{2}
59 1+(3.05−2.22i)T+(18.2−56.1i)T2 1 + (3.05 - 2.22i)T + (18.2 - 56.1i)T^{2}
61 1+(−8.76−6.36i)T+(18.8+58.0i)T2 1 + (-8.76 - 6.36i)T + (18.8 + 58.0i)T^{2}
67 1+(−4.11−1.33i)T+(54.2+39.3i)T2 1 + (-4.11 - 1.33i)T + (54.2 + 39.3i)T^{2}
71 1+(4.09+12.5i)T+(−57.4+41.7i)T2 1 + (4.09 + 12.5i)T + (-57.4 + 41.7i)T^{2}
73 1+(−2.47+3.40i)T+(−22.5−69.4i)T2 1 + (-2.47 + 3.40i)T + (-22.5 - 69.4i)T^{2}
79 1+(−3.05−9.41i)T+(−63.9+46.4i)T2 1 + (-3.05 - 9.41i)T + (-63.9 + 46.4i)T^{2}
83 1+(−4.44−1.44i)T+(67.1+48.7i)T2 1 + (-4.44 - 1.44i)T + (67.1 + 48.7i)T^{2}
89 1+(7.43+5.39i)T+(27.5+84.6i)T2 1 + (7.43 + 5.39i)T + (27.5 + 84.6i)T^{2}
97 1+(0.0857−0.0278i)T+(78.4−57.0i)T2 1 + (0.0857 - 0.0278i)T + (78.4 - 57.0i)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

−12.50168061351087280279261389955, −11.52956187730900840152142668728, −10.69200574474216613745939999307, −9.837985335754167780056587334074, −8.670061287413652810654708878076, −7.83843511878621412530985430844, −6.44677915188911902472949247519, −5.50464523437364720374964653074, −4.92447219441194796258033307435, −2.67508809002860642110573996822, 0.090786021502937786033247959411, 2.07849459477000536661877081971, 4.02066871281858087168298368024, 5.10208969819098859262298256957, 6.72563604207811579598811714659, 7.26294156865133128446426165200, 8.694516595755901209366606227791, 9.783024369479531557725275643299, 10.71183983302168311182645997768, 11.36226191449777862988872221517

Graph of the ZZ-function along the critical line