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Properties

Label 2-975-13.9-c1-0-16
Degree $2$
Conductor $975$
Sign $0.477 - 0.878i$
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  + (−1.13 + 1.95i)2-s + (−0.5 + 0.866i)3-s + (−1.55 − 2.69i)4-s + (−1.13 − 1.95i)6-s + (−0.630 − 1.09i)7-s + 2.52·8-s + (−0.499 − 0.866i)9-s + (−2.26 + 3.91i)11-s + 3.11·12-s + (3.45 − 1.04i)13-s + 2.85·14-s + (0.261 − 0.453i)16-s + (−2.24 − 3.89i)17-s + 2.26·18-s + (−2.55 − 4.43i)19-s + ⋯
L(s)  = 1  + (−0.799 + 1.38i)2-s + (−0.288 + 0.499i)3-s + (−0.778 − 1.34i)4-s + (−0.461 − 0.799i)6-s + (−0.238 − 0.413i)7-s + 0.892·8-s + (−0.166 − 0.288i)9-s + (−0.681 + 1.18i)11-s + 0.899·12-s + (0.957 − 0.290i)13-s + 0.762·14-s + (0.0654 − 0.113i)16-s + (−0.544 − 0.943i)17-s + 0.533·18-s + (−0.586 − 1.01i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593405 + 0.352984i\)
\(L(\frac12)\) \(\approx\) \(0.593405 + 0.352984i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-3.45 + 1.04i)T \)
good2 \( 1 + (1.13 - 1.95i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (0.630 + 1.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.26 - 3.91i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.24 + 3.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.55 + 4.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.11 + 1.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.688 + 1.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.87T + 31T^{2} \)
37 \( 1 + (0.115 - 0.200i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.573 - 0.992i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.18 - 5.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 4.52T + 53T^{2} \)
59 \( 1 + (-0.426 - 0.739i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.31 + 4.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.56 + 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.80 - 8.32i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 + 8.87T + 79T^{2} \)
83 \( 1 - 8.23T + 83T^{2} \)
89 \( 1 + (3.31 - 5.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.33 - 9.24i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.954954827211871348583968227846, −9.183533469397956423768489084954, −8.466870944396675394702955113294, −7.55242711099022616761218024373, −6.81387183130843343386587753667, −6.16102996648918196663144385124, −5.03828640845888125372792219126, −4.39164412266106473488770971934, −2.74211479235186458407271384125, −0.57081323713716158860888550891, 0.969473253422968712456953458433, 2.12735876206708734679709516145, 3.14209728367318471889374288345, 4.11448031775121497596145138084, 5.78698825449160988093884732352, 6.27022754267811948680714068109, 7.75066718777432431371597811462, 8.623971267556518415511317462319, 8.855743622713326469183123570699, 10.23141586681988209658633965348

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