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Properties

Label 2-975-975.116-c0-0-0
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  + (−1.30 − 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 + 1.53i)4-s + (−0.309 + 0.951i)5-s + (0.499 − 1.53i)6-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (1.30 − 0.951i)10-s + (0.5 + 0.363i)11-s + (−1.30 + 0.951i)12-s + (−0.809 + 0.587i)13-s − 0.999·15-s + 1.61·18-s − 1.61·20-s + (−0.309 − 0.951i)22-s + ⋯
L(s)  = 1  + (−1.30 − 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 + 1.53i)4-s + (−0.309 + 0.951i)5-s + (0.499 − 1.53i)6-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (1.30 − 0.951i)10-s + (0.5 + 0.363i)11-s + (−1.30 + 0.951i)12-s + (−0.809 + 0.587i)13-s − 0.999·15-s + 1.61·18-s − 1.61·20-s + (−0.309 − 0.951i)22-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

\(L(\frac{1}{2})\) \(\approx\) \(0.4194252521\)
\(L(\frac12)\) \(\approx\) \(0.4194252521\)
\(L(1)\) 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 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
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

−10.28756438193609838118427465006, −9.725354422578182906246477922690, −9.123866629402467042849405355547, −8.207423080091535180228032391122, −7.46632886436946423246604390912, −6.46517287970465888764526262114, −4.93515675424103969504821623115, −3.80971200878618527509075635769, −2.94069853724917085546255836929, −2.01001582680598952280657595760, 0.58205258744725584290320227253, 1.84532716926272554628085324802, 3.54078499553875350610973579749, 5.16007275678447644454952253989, 5.98660261651157663732442032912, 6.99941495345233610154052873274, 7.51537840027359890570655509687, 8.479384829291889422265239179482, 8.681275063213384475416285694954, 9.599938709831568717994275983407

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