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Properties

Label 2-3150-105.89-c1-0-15
Degree $2$
Conductor $3150$
Sign $-0.641 - 0.766i$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.189 + 2.63i)7-s + 0.999·8-s + (4.67 − 2.69i)11-s − 2.51·13-s + (−2.19 − 1.48i)14-s + (−0.5 + 0.866i)16-s + (−3.89 + 2.24i)17-s + (2.48 + 1.43i)19-s + 5.39i·22-s + (0.133 − 0.232i)23-s + (1.25 − 2.18i)26-s + (2.38 − 1.15i)28-s + 8.89i·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.0716 + 0.997i)7-s + 0.353·8-s + (1.40 − 0.813i)11-s − 0.698·13-s + (−0.585 − 0.396i)14-s + (−0.125 + 0.216i)16-s + (−0.945 + 0.545i)17-s + (0.568 + 0.328i)19-s + 1.15i·22-s + (0.0279 − 0.0483i)23-s + (0.246 − 0.427i)26-s + (0.449 − 0.218i)28-s + 1.65i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.641 - 0.766i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.641 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.197691405\)
\(L(\frac12)\) \(\approx\) \(1.197691405\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.189 - 2.63i)T \)
good11 \( 1 + (-4.67 + 2.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.51T + 13T^{2} \)
17 \( 1 + (3.89 - 2.24i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.48 - 1.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.133 + 0.232i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.89iT - 29T^{2} \)
31 \( 1 + (-4.18 + 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.64 + 3.25i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.760T + 41T^{2} \)
43 \( 1 + 5.86iT - 43T^{2} \)
47 \( 1 + (-6.92 - 3.99i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.19 - 7.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.33 - 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.27 + 1.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.50 - 4.91i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + (-5.82 - 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.29 + 7.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.45iT - 83T^{2} \)
89 \( 1 + (3.98 - 6.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.16T + 97T^{2} \)
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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

−8.977881443971063427617790852089, −8.411640463635229088708348085099, −7.36601445899020130376726644131, −6.71008775591318341882324887617, −5.97930495158618605927721885239, −5.39181683281679351364510880835, −4.39064065550058921074849940302, −3.45880484291004221923772014686, −2.32249445507596185374696160750, −1.17679016284614833623606202881, 0.46644214679692858849003423731, 1.59491326049590042575070228288, 2.59321066645864603867200225054, 3.70419270360435386094816291168, 4.37133869558672098460962550954, 5.01945142542663487638477222700, 6.50291046095338672174993386328, 6.93618710005702024083370574135, 7.65262849589558557033795705236, 8.522573879360836657135535395472

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