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Properties

Label 2-975-195.44-c1-0-14
Degree $2$
Conductor $975$
Sign $-0.627 - 0.778i$
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  + (−0.260 + 0.260i)2-s + (1.26 − 1.18i)3-s + 1.86i·4-s + (−0.0199 + 0.637i)6-s + (−2.54 + 2.54i)7-s + (−1.00 − 1.00i)8-s + (0.187 − 2.99i)9-s + (0.348 − 0.348i)11-s + (2.21 + 2.35i)12-s + (−3.58 + 0.339i)13-s − 1.32i·14-s − 3.20·16-s + 5.28i·17-s + (0.730 + 0.828i)18-s + (3.44 − 3.44i)19-s + ⋯
L(s)  = 1  + (−0.184 + 0.184i)2-s + (0.728 − 0.684i)3-s + 0.932i·4-s + (−0.00814 + 0.260i)6-s + (−0.961 + 0.961i)7-s + (−0.355 − 0.355i)8-s + (0.0625 − 0.998i)9-s + (0.105 − 0.105i)11-s + (0.638 + 0.679i)12-s + (−0.995 + 0.0942i)13-s − 0.354i·14-s − 0.801·16-s + 1.28i·17-s + (0.172 + 0.195i)18-s + (0.791 − 0.791i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.627 - 0.778i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.627 - 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.427489 + 0.894192i\)
\(L(\frac12)\) \(\approx\) \(0.427489 + 0.894192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.26 + 1.18i)T \)
5 \( 1 \)
13 \( 1 + (3.58 - 0.339i)T \)
good2 \( 1 + (0.260 - 0.260i)T - 2iT^{2} \)
7 \( 1 + (2.54 - 2.54i)T - 7iT^{2} \)
11 \( 1 + (-0.348 + 0.348i)T - 11iT^{2} \)
17 \( 1 - 5.28iT - 17T^{2} \)
19 \( 1 + (-3.44 + 3.44i)T - 19iT^{2} \)
23 \( 1 - 9.38iT - 23T^{2} \)
29 \( 1 - 6.80iT - 29T^{2} \)
31 \( 1 + (2.96 - 2.96i)T - 31iT^{2} \)
37 \( 1 + (1.76 - 1.76i)T - 37iT^{2} \)
41 \( 1 + (2.21 + 2.21i)T + 41iT^{2} \)
43 \( 1 + 5.41T + 43T^{2} \)
47 \( 1 + (2.46 + 2.46i)T + 47iT^{2} \)
53 \( 1 + 6.94T + 53T^{2} \)
59 \( 1 + (-0.248 + 0.248i)T - 59iT^{2} \)
61 \( 1 + 3.60T + 61T^{2} \)
67 \( 1 + (1.34 + 1.34i)T + 67iT^{2} \)
71 \( 1 + (-11.6 - 11.6i)T + 71iT^{2} \)
73 \( 1 + (-6.48 + 6.48i)T - 73iT^{2} \)
79 \( 1 + 2.66T + 79T^{2} \)
83 \( 1 + (-7.35 + 7.35i)T - 83iT^{2} \)
89 \( 1 + (-2.54 + 2.54i)T - 89iT^{2} \)
97 \( 1 + (-10.4 - 10.4i)T + 97iT^{2} \)
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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

−9.877925290118208680832097443525, −9.208624809345004215097862984755, −8.709410358020490439857796666842, −7.75875485206838829167032798502, −7.08006585217771661191374752911, −6.36352483415083904950750757071, −5.19997014322778960089414454422, −3.51437251142135333917463273097, −3.13574647700430631836373476124, −1.90718000980439633278754096264, 0.41508029689248784273642622879, 2.23323359429583177762204521616, 3.21849574765612259146719951085, 4.39364537413796011067747267727, 5.12445910428212664486507061377, 6.34698375015123513513827922867, 7.21907558730036405399770036142, 8.120436579660321769859106458111, 9.348689113806093864656412576905, 9.725274871541540778700585289237

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