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Properties

Label 2-315-5.2-c2-0-15
Degree $2$
Conductor $315$
Sign $-0.775 + 0.630i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.24 − 2.24i)2-s + 6.07i·4-s + (3.05 − 3.96i)5-s + (1.87 + 1.87i)7-s + (4.66 − 4.66i)8-s + (−15.7 + 2.04i)10-s − 3.94·11-s + (8.57 − 8.57i)13-s − 8.39i·14-s + 3.37·16-s + (17.2 + 17.2i)17-s − 24.3i·19-s + (24.0 + 18.5i)20-s + (8.85 + 8.85i)22-s + (19.6 − 19.6i)23-s + ⋯
L(s)  = 1  + (−1.12 − 1.12i)2-s + 1.51i·4-s + (0.610 − 0.792i)5-s + (0.267 + 0.267i)7-s + (0.582 − 0.582i)8-s + (−1.57 + 0.204i)10-s − 0.358·11-s + (0.659 − 0.659i)13-s − 0.599i·14-s + 0.210·16-s + (1.01 + 1.01i)17-s − 1.28i·19-s + (1.20 + 0.926i)20-s + (0.402 + 0.402i)22-s + (0.855 − 0.855i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.630i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.775 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.775 + 0.630i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.775 + 0.630i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.321195 - 0.904029i\)
\(L(\frac12)\) \(\approx\) \(0.321195 - 0.904029i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-3.05 + 3.96i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good2 \( 1 + (2.24 + 2.24i)T + 4iT^{2} \)
11 \( 1 + 3.94T + 121T^{2} \)
13 \( 1 + (-8.57 + 8.57i)T - 169iT^{2} \)
17 \( 1 + (-17.2 - 17.2i)T + 289iT^{2} \)
19 \( 1 + 24.3iT - 361T^{2} \)
23 \( 1 + (-19.6 + 19.6i)T - 529iT^{2} \)
29 \( 1 + 17.5iT - 841T^{2} \)
31 \( 1 + 43.8T + 961T^{2} \)
37 \( 1 + (-32.9 - 32.9i)T + 1.36e3iT^{2} \)
41 \( 1 + 22.4T + 1.68e3T^{2} \)
43 \( 1 + (-14.3 + 14.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (38.7 + 38.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (9.01 - 9.01i)T - 2.80e3iT^{2} \)
59 \( 1 + 58.0iT - 3.48e3T^{2} \)
61 \( 1 + 89.2T + 3.72e3T^{2} \)
67 \( 1 + (21.2 + 21.2i)T + 4.48e3iT^{2} \)
71 \( 1 - 78.8T + 5.04e3T^{2} \)
73 \( 1 + (18.2 - 18.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 112. iT - 6.24e3T^{2} \)
83 \( 1 + (-12.9 + 12.9i)T - 6.88e3iT^{2} \)
89 \( 1 + 22.2iT - 7.92e3T^{2} \)
97 \( 1 + (-90.6 - 90.6i)T + 9.40e3iT^{2} \)
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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

−10.85476102806634593643147421596, −10.20359561588565016959048878729, −9.228763935413944656537796651407, −8.573929692594504416628702736729, −7.80092992542651179115122934037, −6.06744650577325763786056832605, −4.94711746658187171143744676282, −3.25327793568397577024113204711, −1.95694041655755578743031717165, −0.73504425478758666926098834193, 1.44303100832962938765896404719, 3.38572826460647518921625845050, 5.36331482302565907157044167894, 6.16365750435398629945347141919, 7.25422076500130296684546231856, 7.73420838011894968901940824708, 9.033523522214212044815137472250, 9.681190818602883494346880971706, 10.56237137369411661361639959216, 11.40605680789024603453046722816

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