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Properties

Label 2-315-7.2-c3-0-24
Degree $2$
Conductor $315$
Sign $0.997 - 0.0706i$
Analytic cond. $18.5856$
Root an. cond. $4.31110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.20 + 3.82i)2-s + (−5.74 − 9.94i)4-s + (2.5 − 4.33i)5-s + (16.1 + 9.14i)7-s + 15.3·8-s + (11.0 + 19.1i)10-s + (−0.0710 − 0.123i)11-s − 32.1·13-s + (−70.4 + 41.3i)14-s + (11.9 − 20.7i)16-s + (−57.1 − 99.0i)17-s + (−21.6 + 37.4i)19-s − 57.4·20-s + 0.627·22-s + (77.2 − 133. i)23-s + ⋯
L(s)  = 1  + (−0.780 + 1.35i)2-s + (−0.717 − 1.24i)4-s + (0.223 − 0.387i)5-s + (0.869 + 0.493i)7-s + 0.679·8-s + (0.348 + 0.604i)10-s + (−0.00194 − 0.00337i)11-s − 0.685·13-s + (−1.34 + 0.790i)14-s + (0.187 − 0.324i)16-s + (−0.815 − 1.41i)17-s + (−0.260 + 0.452i)19-s − 0.642·20-s + 0.00608·22-s + (0.700 − 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0706i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.997 - 0.0706i$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ 0.997 - 0.0706i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9472440182\)
\(L(\frac12)\) \(\approx\) \(0.9472440182\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (-16.1 - 9.14i)T \)
good2 \( 1 + (2.20 - 3.82i)T + (-4 - 6.92i)T^{2} \)
11 \( 1 + (0.0710 + 0.123i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 32.1T + 2.19e3T^{2} \)
17 \( 1 + (57.1 + 99.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (21.6 - 37.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-77.2 + 133. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 40.1T + 2.43e4T^{2} \)
31 \( 1 + (37.7 + 65.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-200. + 346. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 95.4T + 6.89e4T^{2} \)
43 \( 1 + 340.T + 7.95e4T^{2} \)
47 \( 1 + (3.74 - 6.48i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (338. + 586. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-398. - 689. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-378. + 655. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-370. - 641. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 37.0T + 3.57e5T^{2} \)
73 \( 1 + (40.4 + 70.0i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-158. + 274. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 945.T + 5.71e5T^{2} \)
89 \( 1 + (-391. + 678. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 393.T + 9.12e5T^{2} \)
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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

−11.14103703196985393536783713356, −9.859114433470253933548985920237, −9.055436587219445859180930675288, −8.377677908703128933272888992953, −7.46066831299146312961138785360, −6.55615188561094965143472133249, −5.39937106284871767352397645276, −4.67252469003489703620678517368, −2.36609705629017173266905724907, −0.48910899845936316484169528356, 1.26219054865109334599256949077, 2.31092710829467463885395321687, 3.61589708254523937860258677336, 4.86176075914690245373890026475, 6.47515997992451564015200822392, 7.74189716848696919438905657819, 8.612713424287304561189647523062, 9.601252412807648150652974078363, 10.44553953692365210270045563573, 11.08349239063414376506378570833

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