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Properties

Label 2-250-125.106-c1-0-4
Degree $2$
Conductor $250$
Sign $-0.351 - 0.936i$
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  + (−0.535 + 0.844i)2-s + (−0.468 + 0.439i)3-s + (−0.425 − 0.904i)4-s + (1.01 + 1.99i)5-s + (−0.120 − 0.630i)6-s + (1.38 − 1.00i)7-s + (0.992 + 0.125i)8-s + (−0.162 + 2.58i)9-s + (−2.22 − 0.209i)10-s + (−0.894 + 1.40i)11-s + (0.597 + 0.236i)12-s + (0.184 − 2.93i)13-s + (0.107 + 1.70i)14-s + (−1.35 − 0.485i)15-s + (−0.637 + 0.770i)16-s + (−2.96 + 6.29i)17-s + ⋯
L(s)  = 1  + (−0.378 + 0.597i)2-s + (−0.270 + 0.253i)3-s + (−0.212 − 0.452i)4-s + (0.454 + 0.890i)5-s + (−0.0491 − 0.257i)6-s + (0.522 − 0.379i)7-s + (0.350 + 0.0443i)8-s + (−0.0541 + 0.860i)9-s + (−0.703 − 0.0662i)10-s + (−0.269 + 0.424i)11-s + (0.172 + 0.0682i)12-s + (0.0511 − 0.813i)13-s + (0.0286 + 0.456i)14-s + (−0.348 − 0.125i)15-s + (−0.159 + 0.192i)16-s + (−0.718 + 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $-0.351 - 0.936i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ -0.351 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.556649 + 0.803895i\)
\(L(\frac12)\) \(\approx\) \(0.556649 + 0.803895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.535 - 0.844i)T \)
5 \( 1 + (-1.01 - 1.99i)T \)
good3 \( 1 + (0.468 - 0.439i)T + (0.188 - 2.99i)T^{2} \)
7 \( 1 + (-1.38 + 1.00i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.894 - 1.40i)T + (-4.68 - 9.95i)T^{2} \)
13 \( 1 + (-0.184 + 2.93i)T + (-12.8 - 1.62i)T^{2} \)
17 \( 1 + (2.96 - 6.29i)T + (-10.8 - 13.0i)T^{2} \)
19 \( 1 + (-5.16 - 4.85i)T + (1.19 + 18.9i)T^{2} \)
23 \( 1 + (6.29 + 1.61i)T + (20.1 + 11.0i)T^{2} \)
29 \( 1 + (-2.54 - 1.40i)T + (15.5 + 24.4i)T^{2} \)
31 \( 1 + (-2.92 + 6.21i)T + (-19.7 - 23.8i)T^{2} \)
37 \( 1 + (0.290 - 0.351i)T + (-6.93 - 36.3i)T^{2} \)
41 \( 1 + (-8.35 + 2.14i)T + (35.9 - 19.7i)T^{2} \)
43 \( 1 + (-2.21 + 6.83i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-8.69 + 1.09i)T + (45.5 - 11.6i)T^{2} \)
53 \( 1 + (-1.17 + 6.13i)T + (-49.2 - 19.5i)T^{2} \)
59 \( 1 + (0.671 + 0.265i)T + (43.0 + 40.3i)T^{2} \)
61 \( 1 + (11.3 + 2.90i)T + (53.4 + 29.3i)T^{2} \)
67 \( 1 + (-2.63 + 1.45i)T + (35.9 - 56.5i)T^{2} \)
71 \( 1 + (-11.6 + 1.47i)T + (68.7 - 17.6i)T^{2} \)
73 \( 1 + (-10.1 + 4.00i)T + (53.2 - 49.9i)T^{2} \)
79 \( 1 + (-9.26 + 8.70i)T + (4.96 - 78.8i)T^{2} \)
83 \( 1 + (-0.164 - 0.154i)T + (5.21 + 82.8i)T^{2} \)
89 \( 1 + (1.23 - 0.487i)T + (64.8 - 60.9i)T^{2} \)
97 \( 1 + (7.48 + 4.11i)T + (51.9 + 81.8i)T^{2} \)
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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

−12.32117849804249794190507532597, −10.86658645577886393004340212801, −10.51886172709117690481130300799, −9.695517993393406065995071305169, −8.072642662007966344426424947366, −7.64829211102772340943965890628, −6.23573775463882322975049892782, −5.45740504285095073840459734252, −4.05674881070629437645546336898, −2.08687702641133550948176897968, 0.977012954475154459186110367678, 2.61120029788912431236735935515, 4.41490679201876657904264395222, 5.47950441018423860766146456128, 6.78248013808264383711465910038, 8.076767857817075339169486787045, 9.257715605895259464432292124497, 9.435819339840311792558601485175, 11.06484570076251813861452496251, 11.83333791176320875649780308087

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