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Properties

Label 2-975-13.12-c1-0-27
Degree $2$
Conductor $975$
Sign $0.712 - 0.701i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + 0.490i·2-s + 3-s + 1.75·4-s + 0.490i·6-s + 3.59i·7-s + 1.84i·8-s + 9-s − 4.35i·11-s + 1.75·12-s + (2.56 − 2.53i)13-s − 1.75·14-s + 2.61·16-s + 4.13·17-s + 0.490i·18-s − 1.74i·19-s + ⋯
L(s)  = 1  + 0.346i·2-s + 0.577·3-s + 0.879·4-s + 0.200i·6-s + 1.35i·7-s + 0.651i·8-s + 0.333·9-s − 1.31i·11-s + 0.508·12-s + (0.712 − 0.701i)13-s − 0.470·14-s + 0.654·16-s + 1.00·17-s + 0.115i·18-s − 0.401i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.712 - 0.701i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.712 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35513 + 0.965376i\)
\(L(\frac12)\) \(\approx\) \(2.35513 + 0.965376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
13 \( 1 + (-2.56 + 2.53i)T \)
good2 \( 1 - 0.490iT - 2T^{2} \)
7 \( 1 - 3.59iT - 7T^{2} \)
11 \( 1 + 4.35iT - 11T^{2} \)
17 \( 1 - 4.13T + 17T^{2} \)
19 \( 1 + 1.74iT - 19T^{2} \)
23 \( 1 + 0.376T + 23T^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 - 6.90iT - 31T^{2} \)
37 \( 1 - 4.84iT - 37T^{2} \)
41 \( 1 - 4.84iT - 41T^{2} \)
43 \( 1 + 3.24T + 43T^{2} \)
47 \( 1 - 6.13iT - 47T^{2} \)
53 \( 1 + 3.75T + 53T^{2} \)
59 \( 1 + 13.8iT - 59T^{2} \)
61 \( 1 - 8.51T + 61T^{2} \)
67 \( 1 + 1.25iT - 67T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 + 9.41T + 79T^{2} \)
83 \( 1 + 16.8iT - 83T^{2} \)
89 \( 1 + 2.94iT - 89T^{2} \)
97 \( 1 + 10.2iT - 97T^{2} \)
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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

−10.06719352700627594497283171481, −9.026728205487613830573878401337, −8.333444715988221221367529590122, −7.82904568684072569157790267318, −6.61246891754751410681127130749, −5.83756319945862398372874894721, −5.25706363320167939628122922787, −3.34814669084853667083681840853, −2.90359217461147875519540478929, −1.55559296063227462788223600377, 1.32274828871299781872996439143, 2.25280619890136800859266393044, 3.71794194310207854824879810940, 4.08203230302237966033542224585, 5.64588291446288888320048038868, 6.85714684729863027113627167539, 7.32733779685773957088343601979, 7.991449125733700015823269466319, 9.347554166000611289473096622580, 10.00239102394222759187275905225

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