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Properties

Label 2-250-125.104-c1-0-6
Degree $2$
Conductor $250$
Sign $0.508 - 0.861i$
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  + (0.248 + 0.968i)2-s + (3.17 − 0.605i)3-s + (−0.876 + 0.481i)4-s + (−0.921 + 2.03i)5-s + (1.37 + 2.92i)6-s + (−0.889 + 1.22i)7-s + (−0.684 − 0.728i)8-s + (6.90 − 2.73i)9-s + (−2.20 − 0.386i)10-s + (−3.31 + 0.850i)11-s + (−2.48 + 2.05i)12-s + (−0.385 − 0.972i)13-s + (−1.40 − 0.557i)14-s + (−1.69 + 7.01i)15-s + (0.535 − 0.844i)16-s + (2.14 − 3.89i)17-s + ⋯
L(s)  = 1  + (0.175 + 0.684i)2-s + (1.83 − 0.349i)3-s + (−0.438 + 0.240i)4-s + (−0.412 + 0.911i)5-s + (0.561 + 1.19i)6-s + (−0.336 + 0.462i)7-s + (−0.242 − 0.257i)8-s + (2.30 − 0.911i)9-s + (−0.696 − 0.122i)10-s + (−0.998 + 0.256i)11-s + (−0.718 + 0.594i)12-s + (−0.106 − 0.269i)13-s + (−0.376 − 0.148i)14-s + (−0.436 + 1.81i)15-s + (0.133 − 0.211i)16-s + (0.519 − 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $0.508 - 0.861i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ 0.508 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75265 + 1.00073i\)
\(L(\frac12)\) \(\approx\) \(1.75265 + 1.00073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.248 - 0.968i)T \)
5 \( 1 + (0.921 - 2.03i)T \)
good3 \( 1 + (-3.17 + 0.605i)T + (2.78 - 1.10i)T^{2} \)
7 \( 1 + (0.889 - 1.22i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (3.31 - 0.850i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (0.385 + 0.972i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (-2.14 + 3.89i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (-1.18 + 6.20i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-4.71 - 0.296i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (6.25 + 0.789i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (-0.571 - 0.314i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (0.717 + 0.455i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.697 - 11.0i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (2.19 - 0.712i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (7.59 - 8.08i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (6.86 + 3.22i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (0.0573 + 0.0693i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (-0.843 + 13.4i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-0.888 - 7.03i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-7.98 - 7.50i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (9.45 + 7.81i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (-0.997 - 5.22i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (-3.05 - 0.581i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-3.17 + 3.83i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-0.444 + 3.51i)T + (-93.9 - 24.1i)T^{2} \)
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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

−12.78626828809531927335919845409, −11.29598069206983195503951896408, −9.815946896062639278800621059253, −9.204096235506747726713688055657, −7.997087747958637533215352847762, −7.48810447721904400520963158237, −6.63419777987277996074279687036, −4.85396445556270882853961470233, −3.22806907554874087483962258182, −2.69445388693275882640244946878, 1.78050718643118964490466215373, 3.35136752576673296399408898983, 3.97777769994205403224221987348, 5.28462185114839528307375232420, 7.47345044241172862902933603382, 8.231585738010497211789973914326, 8.996954531887873198779607546443, 9.924305691742849896054666951788, 10.63168185860968291609115047064, 12.21140745689764762274844902003

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