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Properties

Label 2-950-95.37-c1-0-19
Degree $2$
Conductor $950$
Sign $0.000346 + 0.999i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (1 + i)7-s + (0.707 + 0.707i)8-s − 3i·9-s + 2·11-s + (−4.24 − 4.24i)13-s − 1.41·14-s − 1.00·16-s + (−3 − 3i)17-s + (2.12 + 2.12i)18-s + (−4.24 − i)19-s + (−1.41 + 1.41i)22-s + (−1 + i)23-s + 6·26-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.377 + 0.377i)7-s + (0.250 + 0.250i)8-s − i·9-s + 0.603·11-s + (−1.17 − 1.17i)13-s − 0.377·14-s − 0.250·16-s + (−0.727 − 0.727i)17-s + (0.499 + 0.499i)18-s + (−0.973 − 0.229i)19-s + (−0.301 + 0.301i)22-s + (−0.208 + 0.208i)23-s + 1.17·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.000346 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.000346 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.000346 + 0.999i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.000346 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515646 - 0.515468i\)
\(L(\frac12)\) \(\approx\) \(0.515646 - 0.515468i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
19 \( 1 + (4.24 + i)T \)
good3 \( 1 + 3iT^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (4.24 + 4.24i)T + 13iT^{2} \)
17 \( 1 + (3 + 3i)T + 17iT^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8.48iT - 31T^{2} \)
37 \( 1 + (4.24 - 4.24i)T - 37iT^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 + (7 + 7i)T + 47iT^{2} \)
53 \( 1 + (4.24 + 4.24i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-8.48 + 8.48i)T - 67iT^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 + (9 - 9i)T - 83iT^{2} \)
89 \( 1 + 8.48T + 89T^{2} \)
97 \( 1 + (-4.24 + 4.24i)T - 97iT^{2} \)
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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

−9.733481843595380708513854051730, −8.856818568140588621294023049191, −8.359304892088372075059695532195, −7.13520518520205712651143016804, −6.67435931421469183212826922364, −5.52304687314432918540824075741, −4.76340144625441209998958967965, −3.40739908760619658949014812453, −2.07534121792859368613321890584, −0.38624097612453340983293759361, 1.68924873085733789322635244842, 2.47943300863781504310087422875, 4.20735447454167606991625537542, 4.54780305985432895085016373007, 6.08870258306279209215369112420, 7.05687493666336234383633393980, 7.85741827901461177151437084858, 8.616319607582732188800536537537, 9.532912882616511718997602048147, 10.19761205049266107457987792822

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