Properties

Label 2-250-125.109-c1-0-1
Degree $2$
Conductor $250$
Sign $-0.639 - 0.768i$
Analytic cond. $1.99626$
Root an. cond. $1.41289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.125 + 0.992i)2-s + (0.477 − 0.394i)3-s + (−0.968 − 0.248i)4-s + (0.393 + 2.20i)5-s + (0.331 + 0.522i)6-s + (−4.08 + 1.32i)7-s + (0.368 − 0.929i)8-s + (−0.490 + 2.57i)9-s + (−2.23 + 0.114i)10-s + (−3.37 − 0.426i)11-s + (−0.560 + 0.263i)12-s + (6.96 + 1.32i)13-s + (−0.805 − 4.22i)14-s + (1.05 + 0.894i)15-s + (0.876 + 0.481i)16-s + (0.922 + 3.59i)17-s + ⋯
L(s)  = 1  + (−0.0886 + 0.701i)2-s + (0.275 − 0.227i)3-s + (−0.484 − 0.124i)4-s + (0.175 + 0.984i)5-s + (0.135 + 0.213i)6-s + (−1.54 + 0.501i)7-s + (0.130 − 0.328i)8-s + (−0.163 + 0.856i)9-s + (−0.706 + 0.0361i)10-s + (−1.01 − 0.128i)11-s + (−0.161 + 0.0761i)12-s + (1.93 + 0.368i)13-s + (−0.215 − 1.12i)14-s + (0.272 + 0.231i)15-s + (0.219 + 0.120i)16-s + (0.223 + 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $-0.639 - 0.768i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ -0.639 - 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418166 + 0.891746i\)
\(L(\frac12)\) \(\approx\) \(0.418166 + 0.891746i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.125 - 0.992i)T \)
5 \( 1 + (-0.393 - 2.20i)T \)
good3 \( 1 + (-0.477 + 0.394i)T + (0.562 - 2.94i)T^{2} \)
7 \( 1 + (4.08 - 1.32i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (3.37 + 0.426i)T + (10.6 + 2.73i)T^{2} \)
13 \( 1 + (-6.96 - 1.32i)T + (12.0 + 4.78i)T^{2} \)
17 \( 1 + (-0.922 - 3.59i)T + (-14.8 + 8.18i)T^{2} \)
19 \( 1 + (-1.60 + 1.94i)T + (-3.56 - 18.6i)T^{2} \)
23 \( 1 + (2.20 - 2.34i)T + (-1.44 - 22.9i)T^{2} \)
29 \( 1 + (0.530 + 8.43i)T + (-28.7 + 3.63i)T^{2} \)
31 \( 1 + (-3.32 + 0.854i)T + (27.1 - 14.9i)T^{2} \)
37 \( 1 + (-2.51 + 4.58i)T + (-19.8 - 31.2i)T^{2} \)
41 \( 1 + (-1.77 + 1.67i)T + (2.57 - 40.9i)T^{2} \)
43 \( 1 + (0.577 + 0.795i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-4.61 - 11.6i)T + (-34.2 + 32.1i)T^{2} \)
53 \( 1 + (-2.06 - 1.31i)T + (22.5 + 47.9i)T^{2} \)
59 \( 1 + (-2.18 - 4.63i)T + (-37.6 + 45.4i)T^{2} \)
61 \( 1 + (1.45 + 1.36i)T + (3.83 + 60.8i)T^{2} \)
67 \( 1 + (-1.72 - 0.108i)T + (66.4 + 8.39i)T^{2} \)
71 \( 1 + (-6.95 + 2.75i)T + (51.7 - 48.6i)T^{2} \)
73 \( 1 + (-8.15 - 3.83i)T + (46.5 + 56.2i)T^{2} \)
79 \( 1 + (7.61 + 9.20i)T + (-14.8 + 77.6i)T^{2} \)
83 \( 1 + (-3.36 - 2.78i)T + (15.5 + 81.5i)T^{2} \)
89 \( 1 + (3.20 - 6.80i)T + (-56.7 - 68.5i)T^{2} \)
97 \( 1 + (-0.0615 + 0.00387i)T + (96.2 - 12.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.80237916604061737126020115227, −11.23623678281429320459538421778, −10.39489702666368796541092684783, −9.453932370280661852662173852334, −8.343443229507906834611604026492, −7.46557024976411172220883367891, −6.19126153301315948157242150801, −5.85457020349812919775738487255, −3.75842236134055684246556634164, −2.57264083531450564185008660525, 0.803502433755116183960274231311, 3.08338978276239687174329402565, 3.88935881544728716892814224219, 5.43702668033193243981816620968, 6.56409022492431885860420867759, 8.185117949856989303234631555458, 9.010555877331210422681462215705, 9.831647613123445113849565684981, 10.51985403041830408157207122484, 11.85770857672870238306193772964

Graph of the $Z$-function along the critical line