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Properties

Label 2-975-195.179-c0-0-1
Degree $2$
Conductor $975$
Sign $0.998 - 0.0471i$
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  + (−0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + (0.866 − 1.5i)7-s + (0.499 − 0.866i)9-s − 0.999i·12-s − i·13-s + (−0.499 − 0.866i)16-s + (1.5 + 0.866i)19-s + 1.73i·21-s + 0.999i·27-s + (0.866 + 1.5i)28-s + (0.499 + 0.866i)36-s + (0.5 + 0.866i)39-s + (0.866 + 0.5i)43-s + (0.866 + 0.499i)48-s + (−1 − 1.73i)49-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + (0.866 − 1.5i)7-s + (0.499 − 0.866i)9-s − 0.999i·12-s − i·13-s + (−0.499 − 0.866i)16-s + (1.5 + 0.866i)19-s + 1.73i·21-s + 0.999i·27-s + (0.866 + 1.5i)28-s + (0.499 + 0.866i)36-s + (0.5 + 0.866i)39-s + (0.866 + 0.5i)43-s + (0.866 + 0.499i)48-s + (−1 − 1.73i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.998 - 0.0471i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.998 - 0.0471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7627733425\)
\(L(\frac12)\) \(\approx\) \(0.7627733425\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + iT \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + 1.73T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−10.18863945354725809606002214696, −9.665138723163920567651939078481, −8.400953683789989162753654423497, −7.63440485442352029590414145920, −7.08196140541923604655411812722, −5.67753303725468628286875505596, −4.85934612011265895961897678510, −4.05243665191458890328968573765, −3.30059615882169244256446176012, −1.01539805814403334687306830811, 1.36425995494544265348673443810, 2.40305759366606209771398654208, 4.41998840821261166179728953322, 5.22582508219415018527254727361, 5.68469944884734289482176611724, 6.62844051042552492416580593578, 7.61408668676333485297886221491, 8.751071500989826797010093530113, 9.275470016418029133599979823455, 10.24098459200135677618807953666

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