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Properties

Label 2-950-95.54-c1-0-1
Degree $2$
Conductor $950$
Sign $0.210 - 0.977i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (0.642 − 0.766i)2-s + (−0.524 − 1.43i)3-s + (−0.173 − 0.984i)4-s + (−1.43 − 0.524i)6-s + (−2.33 + 1.34i)7-s + (−0.866 − 0.500i)8-s + (0.5 − 0.419i)9-s + (−1.59 + 2.75i)11-s + (−1.32 + 0.766i)12-s + (−1.96 + 5.41i)13-s + (−0.467 + 2.65i)14-s + (−0.939 + 0.342i)16-s + (−4.18 + 4.99i)17-s − 0.652i·18-s + (−2.82 − 3.31i)19-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.302 − 0.831i)3-s + (−0.0868 − 0.492i)4-s + (−0.587 − 0.213i)6-s + (−0.882 + 0.509i)7-s + (−0.306 − 0.176i)8-s + (0.166 − 0.139i)9-s + (−0.480 + 0.831i)11-s + (−0.383 + 0.221i)12-s + (−0.546 + 1.50i)13-s + (−0.125 + 0.709i)14-s + (−0.234 + 0.0855i)16-s + (−1.01 + 1.21i)17-s − 0.153i·18-s + (−0.648 − 0.761i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.210 - 0.977i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.210 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324696 + 0.262254i\)
\(L(\frac12)\) \(\approx\) \(0.324696 + 0.262254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
5 \( 1 \)
19 \( 1 + (2.82 + 3.31i)T \)
good3 \( 1 + (0.524 + 1.43i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (2.33 - 1.34i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.59 - 2.75i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.96 - 5.41i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (4.18 - 4.99i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.684 + 0.120i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.16 + 1.81i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.22 + 2.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.36iT - 37T^{2} \)
41 \( 1 + (-0.326 + 0.118i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-5.97 - 1.05i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (5.06 + 6.04i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (8.08 - 1.42i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.439 + 0.368i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.509 - 2.89i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.18 - 3.79i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.46 - 8.32i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (5.39 + 14.8i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (8.51 - 3.10i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-7.34 + 4.23i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.27 - 2.64i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (0.223 - 0.266i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.23236472027636352718257303516, −9.487884544056351424638412807266, −8.757974318547441640394656514628, −7.42206891033262871537674360364, −6.53791482161906417255272806954, −6.23897272629997045435656214676, −4.78359010551551000583054303627, −4.03791499029353404931498900867, −2.52653001240779259464581049072, −1.78452099702694801882761329097, 0.16169485516151435809572745482, 2.78086464352410073454881421983, 3.63765337252974086091034799398, 4.69062250940657415384927466130, 5.39675306009225838995592239607, 6.29086796273762637890797840732, 7.27229735088048657522522354299, 8.028536048000575748634120231173, 9.108864731414429690340226498153, 9.956752438952124036591307603497

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