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Properties

Label 2-975-39.35-c0-0-1
Degree $2$
Conductor $975$
Sign $-0.711 + 0.702i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s − 0.999·12-s − 13-s + (−0.499 + 0.866i)16-s + (0.5 + 0.866i)19-s − 0.999·21-s − 0.999·27-s + (−0.499 + 0.866i)28-s + 2·31-s + (−0.499 + 0.866i)36-s + (1 − 1.73i)37-s + (−0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s − 0.999·12-s − 13-s + (−0.499 + 0.866i)16-s + (0.5 + 0.866i)19-s − 0.999·21-s − 0.999·27-s + (−0.499 + 0.866i)28-s + 2·31-s + (−0.499 + 0.866i)36-s + (1 − 1.73i)37-s + (−0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.711 + 0.702i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (776, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8829375268\)
\(L(\frac12)\) \(\approx\) \(0.8829375268\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 2T + T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.852246864281772762571094234387, −9.184768914145084891096575728697, −8.143038880934048142478579951979, −7.37145874042251345594491716672, −6.54631896126174916189396117275, −5.73813529112427156566665246733, −4.57442467153360372299369917498, −3.51420551010126304392432655651, −2.19207251486768035259803088450, −0.813275479504411650719544841679, 2.70408263627851848195748200723, 3.04054851144198347176414363508, 4.46368752189094048442954809538, 4.92930580491230472922330850948, 6.20060069090052131309275776188, 7.41785223496469855228093816686, 8.223345515667233378018295425003, 8.926594045490684485566430927972, 9.610824892434926778972806506377, 10.14481120059794087714542727938

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