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Properties

Label 2-250-125.109-c1-0-3
Degree $2$
Conductor $250$
Sign $0.387 - 0.921i$
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 + (−0.590 + 0.488i)3-s + (−0.968 − 0.248i)4-s + (2.11 − 0.738i)5-s + (−0.410 − 0.646i)6-s + (1.84 − 0.599i)7-s + (0.368 − 0.929i)8-s + (−0.452 + 2.37i)9-s + (0.468 + 2.18i)10-s + (1.66 + 0.210i)11-s + (0.692 − 0.326i)12-s + (1.97 + 0.376i)13-s + (0.363 + 1.90i)14-s + (−0.884 + 1.46i)15-s + (0.876 + 0.481i)16-s + (0.0918 + 0.357i)17-s + ⋯
L(s)  = 1  + (−0.0886 + 0.701i)2-s + (−0.340 + 0.281i)3-s + (−0.484 − 0.124i)4-s + (0.943 − 0.330i)5-s + (−0.167 − 0.263i)6-s + (0.697 − 0.226i)7-s + (0.130 − 0.328i)8-s + (−0.150 + 0.790i)9-s + (0.147 + 0.691i)10-s + (0.502 + 0.0635i)11-s + (0.200 − 0.0941i)12-s + (0.547 + 0.104i)13-s + (0.0972 + 0.509i)14-s + (−0.228 + 0.378i)15-s + (0.219 + 0.120i)16-s + (0.0222 + 0.0867i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $0.387 - 0.921i$
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: \((2,\ 250,\ (\ :1/2),\ 0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05711 + 0.702399i\)
\(L(\frac12)\) \(\approx\) \(1.05711 + 0.702399i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 + 0.738i)T \)
good3 \( 1 + (0.590 - 0.488i)T + (0.562 - 2.94i)T^{2} \)
7 \( 1 + (-1.84 + 0.599i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.66 - 0.210i)T + (10.6 + 2.73i)T^{2} \)
13 \( 1 + (-1.97 - 0.376i)T + (12.0 + 4.78i)T^{2} \)
17 \( 1 + (-0.0918 - 0.357i)T + (-14.8 + 8.18i)T^{2} \)
19 \( 1 + (2.74 - 3.32i)T + (-3.56 - 18.6i)T^{2} \)
23 \( 1 + (0.448 - 0.477i)T + (-1.44 - 22.9i)T^{2} \)
29 \( 1 + (-0.457 - 7.27i)T + (-28.7 + 3.63i)T^{2} \)
31 \( 1 + (4.12 - 1.05i)T + (27.1 - 14.9i)T^{2} \)
37 \( 1 + (-5.31 + 9.67i)T + (-19.8 - 31.2i)T^{2} \)
41 \( 1 + (-7.11 + 6.68i)T + (2.57 - 40.9i)T^{2} \)
43 \( 1 + (6.96 + 9.58i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.492 + 1.24i)T + (-34.2 + 32.1i)T^{2} \)
53 \( 1 + (-1.90 - 1.20i)T + (22.5 + 47.9i)T^{2} \)
59 \( 1 + (2.38 + 5.06i)T + (-37.6 + 45.4i)T^{2} \)
61 \( 1 + (-3.74 - 3.51i)T + (3.83 + 60.8i)T^{2} \)
67 \( 1 + (6.14 + 0.386i)T + (66.4 + 8.39i)T^{2} \)
71 \( 1 + (1.92 - 0.764i)T + (51.7 - 48.6i)T^{2} \)
73 \( 1 + (5.67 + 2.67i)T + (46.5 + 56.2i)T^{2} \)
79 \( 1 + (7.04 + 8.51i)T + (-14.8 + 77.6i)T^{2} \)
83 \( 1 + (0.0795 + 0.0658i)T + (15.5 + 81.5i)T^{2} \)
89 \( 1 + (2.63 - 5.60i)T + (-56.7 - 68.5i)T^{2} \)
97 \( 1 + (10.4 - 0.655i)T + (96.2 - 12.1i)T^{2} \)
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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

−12.37034082617193123854057535709, −10.98274339321183794861317729042, −10.39072839683343236130047756171, −9.181681649531646346848353482299, −8.406928963749424644043442313167, −7.24557213925012359443708767313, −5.97649298808256911743399406255, −5.25757492200340723749113244317, −4.12943427775003945705671494956, −1.76784657765921516503872548308, 1.38600530763259329063452887740, 2.85876208882558963184427465855, 4.45270161384328611841128725747, 5.84442780185611995502680596180, 6.62642423907140849149285891501, 8.182347111543978912039029793647, 9.207400846691308310756594309304, 9.967027885253257114514825722833, 11.25653578581266598731666116587, 11.54530507124104074187000383193

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