Properties

Label 2-950-95.44-c1-0-25
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: $\chi_{950} (899, \cdot )$
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

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

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