Properties

Label 2-950-5.4-c1-0-20
Degree $2$
Conductor $950$
Sign $0.894 + 0.447i$
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  + i·2-s + 3i·3-s − 4-s − 3·6-s − 5i·7-s i·8-s − 6·9-s − 4·11-s − 3i·12-s + i·13-s + 5·14-s + 16-s − 3i·17-s − 6i·18-s − 19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.73i·3-s − 0.5·4-s − 1.22·6-s − 1.88i·7-s − 0.353i·8-s − 2·9-s − 1.20·11-s − 0.866i·12-s + 0.277i·13-s + 1.33·14-s + 0.250·16-s − 0.727i·17-s − 1.41i·18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.556556 - 0.131385i\)
\(L(\frac12)\) \(\approx\) \(0.556556 - 0.131385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 3iT - 3T^{2} \)
7 \( 1 + 5iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
23 \( 1 + 7iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 13iT - 53T^{2} \)
59 \( 1 - 9T + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−10.13005754909113889890701048623, −9.244549850815312309721075645447, −8.340239925267233839920725068411, −7.48816219465705187636854610825, −6.59378533420009187135711703750, −5.33685527201188894107005962850, −4.59185789638246659387353467167, −4.06614574231571688955308805184, −2.98335211476737464017735058856, −0.25849936757337310997818913150, 1.60717875927706914596224029356, 2.39703959808082716387771161589, 3.17293555578459101267045758585, 5.21063747931156207296261021631, 5.72012460110250655705893751189, 6.64126552377188817121822222429, 7.900926595129397868586337442323, 8.267657670479892378188304515591, 9.093704631429224472444185127512, 10.15016545704303514162933428331

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