Properties

Label 2-315-1.1-c7-0-35
Degree $2$
Conductor $315$
Sign $-1$
Analytic cond. $98.4012$
Root an. cond. $9.91974$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.6·2-s + 86.1·4-s − 125·5-s − 343·7-s + 612.·8-s + 1.82e3·10-s + 6.47e3·11-s − 1.16e4·13-s + 5.01e3·14-s − 1.99e4·16-s − 1.34e4·17-s + 3.49e4·19-s − 1.07e4·20-s − 9.47e4·22-s − 7.78e4·23-s + 1.56e4·25-s + 1.70e5·26-s − 2.95e4·28-s + 2.21e5·29-s − 2.32e4·31-s + 2.14e5·32-s + 1.96e5·34-s + 4.28e4·35-s − 4.22e5·37-s − 5.11e5·38-s − 7.65e4·40-s − 1.91e5·41-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.672·4-s − 0.447·5-s − 0.377·7-s + 0.423·8-s + 0.578·10-s + 1.46·11-s − 1.47·13-s + 0.488·14-s − 1.22·16-s − 0.664·17-s + 1.16·19-s − 0.300·20-s − 1.89·22-s − 1.33·23-s + 0.199·25-s + 1.90·26-s − 0.254·28-s + 1.68·29-s − 0.140·31-s + 1.15·32-s + 0.859·34-s + 0.169·35-s − 1.37·37-s − 1.51·38-s − 0.189·40-s − 0.434·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(98.4012\)
Root analytic conductor: \(9.91974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 315,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + 125T \)
7 \( 1 + 343T \)
good2 \( 1 + 14.6T + 128T^{2} \)
11 \( 1 - 6.47e3T + 1.94e7T^{2} \)
13 \( 1 + 1.16e4T + 6.27e7T^{2} \)
17 \( 1 + 1.34e4T + 4.10e8T^{2} \)
19 \( 1 - 3.49e4T + 8.93e8T^{2} \)
23 \( 1 + 7.78e4T + 3.40e9T^{2} \)
29 \( 1 - 2.21e5T + 1.72e10T^{2} \)
31 \( 1 + 2.32e4T + 2.75e10T^{2} \)
37 \( 1 + 4.22e5T + 9.49e10T^{2} \)
41 \( 1 + 1.91e5T + 1.94e11T^{2} \)
43 \( 1 - 3.10e5T + 2.71e11T^{2} \)
47 \( 1 - 2.40e5T + 5.06e11T^{2} \)
53 \( 1 - 1.06e6T + 1.17e12T^{2} \)
59 \( 1 + 4.51e5T + 2.48e12T^{2} \)
61 \( 1 + 8.31e5T + 3.14e12T^{2} \)
67 \( 1 - 2.26e6T + 6.06e12T^{2} \)
71 \( 1 - 2.22e6T + 9.09e12T^{2} \)
73 \( 1 - 4.99e6T + 1.10e13T^{2} \)
79 \( 1 + 2.72e6T + 1.92e13T^{2} \)
83 \( 1 - 6.38e6T + 2.71e13T^{2} \)
89 \( 1 - 7.32e6T + 4.42e13T^{2} \)
97 \( 1 + 2.38e6T + 8.07e13T^{2} \)
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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

−9.762026681107792349655995821107, −9.162688733891007829443087212879, −8.218127872632954978929908046839, −7.25774702023086310329099941795, −6.55973576397895980395111533977, −4.89929784950515326110686431240, −3.80193208114921394573342180703, −2.27866657053432208330529432132, −1.01368303051723173992798088275, 0, 1.01368303051723173992798088275, 2.27866657053432208330529432132, 3.80193208114921394573342180703, 4.89929784950515326110686431240, 6.55973576397895980395111533977, 7.25774702023086310329099941795, 8.218127872632954978929908046839, 9.162688733891007829443087212879, 9.762026681107792349655995821107

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