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Properties

Label 2-315-7.2-c3-0-1
Degree $2$
Conductor $315$
Sign $-0.727 + 0.685i$
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  + (−0.975 + 1.68i)2-s + (2.09 + 3.63i)4-s + (2.5 − 4.33i)5-s + (18.2 − 3.10i)7-s − 23.7·8-s + (4.87 + 8.44i)10-s + (−31.1 − 53.8i)11-s − 87.5·13-s + (−12.5 + 33.8i)14-s + (6.42 − 11.1i)16-s + (20.9 + 36.2i)17-s + (−58.2 + 100. i)19-s + 20.9·20-s + 121.·22-s + (−12.8 + 22.2i)23-s + ⋯
L(s)  = 1  + (−0.344 + 0.597i)2-s + (0.262 + 0.454i)4-s + (0.223 − 0.387i)5-s + (0.985 − 0.167i)7-s − 1.05·8-s + (0.154 + 0.267i)10-s + (−0.852 − 1.47i)11-s − 1.86·13-s + (−0.239 + 0.646i)14-s + (0.100 − 0.173i)16-s + (0.298 + 0.517i)17-s + (−0.703 + 1.21i)19-s + 0.234·20-s + 1.17·22-s + (−0.116 + 0.201i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.727 + 0.685i$
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: \((2,\ 315,\ (\ :3/2),\ -0.727 + 0.685i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.05711528963\)
\(L(\frac12)\) \(\approx\) \(0.05711528963\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.2 + 3.10i)T \)
good2 \( 1 + (0.975 - 1.68i)T + (-4 - 6.92i)T^{2} \)
11 \( 1 + (31.1 + 53.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 87.5T + 2.19e3T^{2} \)
17 \( 1 + (-20.9 - 36.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (58.2 - 100. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (12.8 - 22.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 277.T + 2.43e4T^{2} \)
31 \( 1 + (-61.5 - 106. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (63.5 - 110. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 336.T + 6.89e4T^{2} \)
43 \( 1 + 124.T + 7.95e4T^{2} \)
47 \( 1 + (-113. + 197. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-100. - 173. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (190. + 329. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-233. + 404. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-144. - 250. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 60.3T + 3.57e5T^{2} \)
73 \( 1 + (398. + 689. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (338. - 585. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 966.T + 5.71e5T^{2} \)
89 \( 1 + (248. - 430. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.01e3T + 9.12e5T^{2} \)
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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

−11.87312067921172430141204679234, −10.86007050251173468947891609276, −9.868766779613330239650182845555, −8.557960072294287949567325184663, −8.079872628486427085571123360983, −7.26101363612536339700095653829, −5.90159229985069027397763619662, −5.07320984365798047271077756787, −3.49279676889851857653162427800, −2.04232019703296850596437156294, 0.02048388196186103761285853913, 2.02282503963253557166974843891, 2.52700768009795517540841903338, 4.69403218754262221467514200910, 5.41162752328092096682078054698, 6.98908273311284618868527256349, 7.61281404724084202409207811490, 9.116166265571594848363724882037, 9.905146031582017784019293124755, 10.54129849221023824189196829492

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