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Properties

Label 2-975-65.4-c1-0-34
Degree $2$
Conductor $975$
Sign $-0.668 + 0.743i$
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  + (−0.866 − 0.5i)3-s + (1 + 1.73i)4-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (−3 − 1.73i)11-s − 1.99i·12-s + (−0.866 − 3.5i)13-s + (−1.99 + 3.46i)16-s + (−3 + 1.73i)19-s + 1.73i·21-s + (−5.19 − 3i)23-s − 0.999i·27-s + (1.73 − 3i)28-s + (3 − 5.19i)29-s + 1.73i·31-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.5 + 0.866i)4-s + (−0.327 − 0.566i)7-s + (0.166 + 0.288i)9-s + (−0.904 − 0.522i)11-s − 0.577i·12-s + (−0.240 − 0.970i)13-s + (−0.499 + 0.866i)16-s + (−0.688 + 0.397i)19-s + 0.377i·21-s + (−1.08 − 0.625i)23-s − 0.192i·27-s + (0.327 − 0.566i)28-s + (0.557 − 0.964i)29-s + 0.311i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.668 + 0.743i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.216425 - 0.485328i\)
\(L(\frac12)\) \(\approx\) \(0.216425 - 0.485328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (0.866 + 3.5i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
7 \( 1 + (0.866 + 1.5i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6 + 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + (-3 + 1.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.33 - 7.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (9 - 5.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73T + 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.59 + 4.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.15514191703449419107204996528, −8.439696552862817874908880417330, −8.072838832619015141834590076150, −7.16895125569203056043757436372, −6.37898747648909379785885694321, −5.50160443466166500766676317078, −4.26519291596070556056526880526, −3.24432837146728915081733500909, −2.19711901828591363208333192318, −0.23868351463079220550670169246, 1.73126899006266032750962590832, 2.77925961243309257875682890949, 4.37617964959917134369768447189, 5.17314492709289361203201624034, 6.03302117451393963911737740238, 6.71398312399790807834175908936, 7.63397554869981517023222660514, 8.908531604119325067103182645744, 9.647717554932440854265447744685, 10.32265875601319363859133869127

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