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Properties

Label 2-950-95.37-c1-0-7
Degree $2$
Conductor $950$
Sign $0.765 - 0.643i$
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 + (0.776 + 0.776i)3-s − 1.00i·4-s − 1.09·6-s + (−0.710 − 0.710i)7-s + (0.707 + 0.707i)8-s − 1.79i·9-s − 0.677·11-s + (0.776 − 0.776i)12-s + (1.71 + 1.71i)13-s + 1.00·14-s − 1.00·16-s + (0.103 + 0.103i)17-s + (1.26 + 1.26i)18-s + (1.96 + 3.89i)19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.448 + 0.448i)3-s − 0.500i·4-s − 0.448·6-s + (−0.268 − 0.268i)7-s + (0.250 + 0.250i)8-s − 0.597i·9-s − 0.204·11-s + (0.224 − 0.224i)12-s + (0.474 + 0.474i)13-s + 0.268·14-s − 0.250·16-s + (0.0250 + 0.0250i)17-s + (0.298 + 0.298i)18-s + (0.450 + 0.892i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.765 - 0.643i$
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: \((2,\ 950,\ (\ :1/2),\ 0.765 - 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34515 + 0.490154i\)
\(L(\frac12)\) \(\approx\) \(1.34515 + 0.490154i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.96 - 3.89i)T \)
good3 \( 1 + (-0.776 - 0.776i)T + 3iT^{2} \)
7 \( 1 + (0.710 + 0.710i)T + 7iT^{2} \)
11 \( 1 + 0.677T + 11T^{2} \)
13 \( 1 + (-1.71 - 1.71i)T + 13iT^{2} \)
17 \( 1 + (-0.103 - 0.103i)T + 17iT^{2} \)
23 \( 1 + (-3.40 + 3.40i)T - 23iT^{2} \)
29 \( 1 - 9.60T + 29T^{2} \)
31 \( 1 - 4.78iT - 31T^{2} \)
37 \( 1 + (-5.51 + 5.51i)T - 37iT^{2} \)
41 \( 1 - 5.92iT - 41T^{2} \)
43 \( 1 + (-1.28 + 1.28i)T - 43iT^{2} \)
47 \( 1 + (-3.99 - 3.99i)T + 47iT^{2} \)
53 \( 1 + (-1.10 - 1.10i)T + 53iT^{2} \)
59 \( 1 - 7.60T + 59T^{2} \)
61 \( 1 + 6.27T + 61T^{2} \)
67 \( 1 + (1.59 - 1.59i)T - 67iT^{2} \)
71 \( 1 + 5.83iT - 71T^{2} \)
73 \( 1 + (-1.31 + 1.31i)T - 73iT^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 + (-2.09 + 2.09i)T - 83iT^{2} \)
89 \( 1 - 9.17T + 89T^{2} \)
97 \( 1 + (-0.694 + 0.694i)T - 97iT^{2} \)
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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

−10.06670634671830253039782171729, −9.149068459904740962488229100432, −8.628169527877417311980996379784, −7.72193304004544027566426142516, −6.71030419884975653149133033158, −6.09101163876106365099078886578, −4.83141249320755943441263614107, −3.86293153617685329400786654674, −2.78421158124196119074237182682, −1.04140932545583248592559609420, 1.05327619225771313318315586446, 2.47104696097471527830801760526, 3.12596085336412465055016510381, 4.53926611861172749558047512116, 5.61482658172863590470338549721, 6.81242666455098991309130361770, 7.61551584811651389193609869555, 8.356784388113115952410082795808, 9.045977539505613618391864847449, 9.939319282773241143024064045702

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