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Properties

Label 2-315-315.184-c1-0-37
Degree $2$
Conductor $315$
Sign $-0.154 + 0.987i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  − i·2-s + (0.866 − 1.5i)3-s + 4-s + (1.23 − 1.86i)5-s + (−1.5 − 0.866i)6-s + (1.73 + 2i)7-s − 3i·8-s + (−1.5 − 2.59i)9-s + (−1.86 − 1.23i)10-s + (−3 + 5.19i)11-s + (0.866 − 1.5i)12-s + (3.46 + 2i)13-s + (2 − 1.73i)14-s + (−1.73 − 3.46i)15-s − 16-s + (−1.73 + i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.499 − 0.866i)3-s + 0.5·4-s + (0.550 − 0.834i)5-s + (−0.612 − 0.353i)6-s + (0.654 + 0.755i)7-s − 1.06i·8-s + (−0.5 − 0.866i)9-s + (−0.590 − 0.389i)10-s + (−0.904 + 1.56i)11-s + (0.249 − 0.433i)12-s + (0.960 + 0.554i)13-s + (0.534 − 0.462i)14-s + (−0.447 − 0.894i)15-s − 0.250·16-s + (−0.420 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.154 + 0.987i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.154 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24395 - 1.45417i\)
\(L(\frac12)\) \(\approx\) \(1.24395 - 1.45417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 + (-1.23 + 1.86i)T \)
7 \( 1 + (-1.73 - 2i)T \)
good2 \( 1 + iT - 2T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.46 - 2i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + (-10.3 + 6i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + (-3.46 + 2i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.69935002554589097258048207873, −10.52372963461625026099115574061, −9.534423527478127240900865085099, −8.584990576339520760347291568601, −7.71623143776770987509794827095, −6.53827712553479990005753086006, −5.50209002240075575478134010520, −3.95470157567648578993265603997, −2.01086526851167627854172972250, −1.89691053359203334930380492168, 2.43958974720202154325717235390, 3.51415861428821133978789082534, 5.13920103323531374538216254764, 5.99898790889188780519397328454, 7.12801452526628138689754102129, 8.163469866028281169255793670101, 8.790274186046036163904704058804, 10.41644771011936206496956492978, 10.85656386392846847850766462574, 11.27515883936422165280227934519

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