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Properties

Label 2-975-195.164-c1-0-23
Degree $2$
Conductor $975$
Sign $0.999 + 0.0353i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.999 + 0.0353i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.999 + 0.0353i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59343 - 0.0281986i\)
\(L(\frac12)\) \(\approx\) \(1.59343 - 0.0281986i\)
\(L(\frac{3}{2})\) 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

−10.12937814103678226156671479108, −9.181489253366667300208679490382, −8.144666239153758469053049805706, −7.48267102997051532742343913339, −6.17619059707629267102293762096, −5.81831024854494747182085102128, −5.14850796568036607499230045022, −3.94490586298953331883202689726, −1.97990391361922821300014180654, −1.32786290338138020833572960759, 0.917554132151902709132214941378, 2.85379835260156078712577784125, 3.92406375670245110177616454807, 4.60005087667990427000247430672, 5.36981969414621939617242821685, 6.80302167636589589624760126180, 7.32005662331772570474309605804, 8.496589169054998328836603627223, 9.070731204311007945479590448692, 10.30309322118532111137360913588

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