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Properties

Label 2-315-7.6-c4-0-5
Degree $2$
Conductor $315$
Sign $-0.400 - 0.916i$
Analytic cond. $32.5615$
Root an. cond. $5.70627$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.81·2-s + 7.21·4-s − 11.1i·5-s + (44.8 − 19.6i)7-s + 42.3·8-s + 53.8i·10-s − 201.·11-s + 258. i·13-s + (−216. + 94.5i)14-s − 319.·16-s − 242. i·17-s − 253. i·19-s − 80.6i·20-s + 971.·22-s + 323.·23-s + ⋯
L(s)  = 1  − 1.20·2-s + 0.450·4-s − 0.447i·5-s + (0.916 − 0.400i)7-s + 0.661·8-s + 0.538i·10-s − 1.66·11-s + 1.53i·13-s + (−1.10 + 0.482i)14-s − 1.24·16-s − 0.840i·17-s − 0.702i·19-s − 0.201i·20-s + 2.00·22-s + 0.611·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.400 - 0.916i$
Analytic conductor: \(32.5615\)
Root analytic conductor: \(5.70627\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :2),\ -0.400 - 0.916i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.3300326922\)
\(L(\frac12)\) \(\approx\) \(0.3300326922\)
\(L(3)\) 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 \( 1 + 11.1iT \)
7 \( 1 + (-44.8 + 19.6i)T \)
good2 \( 1 + 4.81T + 16T^{2} \)
11 \( 1 + 201.T + 1.46e4T^{2} \)
13 \( 1 - 258. iT - 2.85e4T^{2} \)
17 \( 1 + 242. iT - 8.35e4T^{2} \)
19 \( 1 + 253. iT - 1.30e5T^{2} \)
23 \( 1 - 323.T + 2.79e5T^{2} \)
29 \( 1 - 331.T + 7.07e5T^{2} \)
31 \( 1 + 278. iT - 9.23e5T^{2} \)
37 \( 1 + 322.T + 1.87e6T^{2} \)
41 \( 1 - 578. iT - 2.82e6T^{2} \)
43 \( 1 + 1.52e3T + 3.41e6T^{2} \)
47 \( 1 + 389. iT - 4.87e6T^{2} \)
53 \( 1 + 1.07e3T + 7.89e6T^{2} \)
59 \( 1 - 1.03e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.68e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.05e3T + 2.01e7T^{2} \)
71 \( 1 + 3.21e3T + 2.54e7T^{2} \)
73 \( 1 + 2.23e3iT - 2.83e7T^{2} \)
79 \( 1 + 5.23e3T + 3.89e7T^{2} \)
83 \( 1 - 3.79e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.36e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.85e4iT - 8.85e7T^{2} \)
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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

−11.07653700868255550373001397398, −10.29477554805920949520050072937, −9.305280137983979747154195224528, −8.568322925199897631915257857691, −7.70199616939361448260441079980, −6.96721713419382361344352212164, −5.11578980855181586404854037318, −4.47469569731395990262688778395, −2.37299317970280262369136743145, −1.11876956778067430116684424521, 0.17214833683557155094410176309, 1.70015030985860450185350253613, 3.00524910201109418739069264785, 4.82801655431818895863692082892, 5.76029624005440861135680416683, 7.38152292458815892454065105826, 8.081095162289623991844581092879, 8.550531291437173431327347418705, 10.01325177736646107227201938070, 10.51610580767627678146576097179

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