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Properties

Label 2-975-975.116-c0-0-1
Degree $2$
Conductor $975$
Sign $0.187 + 0.982i$
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  + (−0.951 − 0.690i)2-s + (−0.309 − 0.951i)3-s + (0.118 + 0.363i)4-s + (0.951 + 0.309i)5-s + (−0.363 + 1.11i)6-s + (−0.224 + 0.690i)8-s + (−0.809 + 0.587i)9-s + (−0.690 − 0.951i)10-s + (1.53 + 1.11i)11-s + (0.309 − 0.224i)12-s + (0.809 − 0.587i)13-s − 0.999i·15-s + (0.999 − 0.726i)16-s + 1.17·18-s + 0.381i·20-s + ⋯
L(s)  = 1  + (−0.951 − 0.690i)2-s + (−0.309 − 0.951i)3-s + (0.118 + 0.363i)4-s + (0.951 + 0.309i)5-s + (−0.363 + 1.11i)6-s + (−0.224 + 0.690i)8-s + (−0.809 + 0.587i)9-s + (−0.690 − 0.951i)10-s + (1.53 + 1.11i)11-s + (0.309 − 0.224i)12-s + (0.809 − 0.587i)13-s − 0.999i·15-s + (0.999 − 0.726i)16-s + 1.17·18-s + 0.381i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6978986337\)
\(L(\frac12)\) \(\approx\) \(0.6978986337\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−10.15130048656278873246480578141, −9.187519099665838777789871241309, −8.662391068379611214501907989646, −7.57018469813874030005077076453, −6.62705165249217629925537279215, −6.00292694668746981601214498290, −4.95176579690773798779778696392, −3.20604889606502990008585571583, −1.93573113953617233586486404030, −1.35411614808421668490715251114, 1.26014343607666632295142841999, 3.33702350373953929706794056454, 4.17279975309835426058644157231, 5.49362029210490437235635819368, 6.30965984000648481319815759399, 6.73591455773996825008701273135, 8.405292255882194890578332796676, 8.764398105370697388173750587383, 9.422321651809056514253275399170, 10.00731519157356376055912943186

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