Properties

Label 2-315-1.1-c3-0-19
Degree $2$
Conductor $315$
Sign $-1$
Analytic cond. $18.5856$
Root an. cond. $4.31110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.70·2-s + 14.1·4-s + 5·5-s + 7·7-s − 28.7·8-s − 23.5·10-s − 24.5·11-s − 35.0·13-s − 32.9·14-s + 22.1·16-s + 18.4·17-s − 67.4·19-s + 70.5·20-s + 115.·22-s + 145.·23-s + 25·25-s + 164.·26-s + 98.7·28-s − 214.·29-s − 88.6·31-s + 125.·32-s − 86.5·34-s + 35·35-s + 162.·37-s + 316.·38-s − 143.·40-s + 337.·41-s + ⋯
L(s)  = 1  − 1.66·2-s + 1.76·4-s + 0.447·5-s + 0.377·7-s − 1.26·8-s − 0.743·10-s − 0.674·11-s − 0.747·13-s − 0.628·14-s + 0.345·16-s + 0.262·17-s − 0.813·19-s + 0.788·20-s + 1.12·22-s + 1.32·23-s + 0.200·25-s + 1.24·26-s + 0.666·28-s − 1.37·29-s − 0.513·31-s + 0.694·32-s − 0.436·34-s + 0.169·35-s + 0.720·37-s + 1.35·38-s − 0.567·40-s + 1.28·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 5T \)
7 \( 1 - 7T \)
good2 \( 1 + 4.70T + 8T^{2} \)
11 \( 1 + 24.5T + 1.33e3T^{2} \)
13 \( 1 + 35.0T + 2.19e3T^{2} \)
17 \( 1 - 18.4T + 4.91e3T^{2} \)
19 \( 1 + 67.4T + 6.85e3T^{2} \)
23 \( 1 - 145.T + 1.21e4T^{2} \)
29 \( 1 + 214.T + 2.43e4T^{2} \)
31 \( 1 + 88.6T + 2.97e4T^{2} \)
37 \( 1 - 162.T + 5.06e4T^{2} \)
41 \( 1 - 337.T + 6.89e4T^{2} \)
43 \( 1 - 122.T + 7.95e4T^{2} \)
47 \( 1 + 354.T + 1.03e5T^{2} \)
53 \( 1 + 676.T + 1.48e5T^{2} \)
59 \( 1 + 501.T + 2.05e5T^{2} \)
61 \( 1 + 708.T + 2.26e5T^{2} \)
67 \( 1 + 907.T + 3.00e5T^{2} \)
71 \( 1 + 430.T + 3.57e5T^{2} \)
73 \( 1 - 41.3T + 3.89e5T^{2} \)
79 \( 1 - 890.T + 4.93e5T^{2} \)
83 \( 1 - 1.05e3T + 5.71e5T^{2} \)
89 \( 1 + 1.47e3T + 7.04e5T^{2} \)
97 \( 1 - 555.T + 9.12e5T^{2} \)
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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

−10.73367796985506401003902973602, −9.589166291319292280483701314920, −9.088519870035481934270409011195, −7.919320767137666454790690186965, −7.34645259867073888196785873983, −6.11845432029063796772736407004, −4.79606338446240584283964420960, −2.72285383104770266038427571937, −1.56122235597130864247503891068, 0, 1.56122235597130864247503891068, 2.72285383104770266038427571937, 4.79606338446240584283964420960, 6.11845432029063796772736407004, 7.34645259867073888196785873983, 7.919320767137666454790690186965, 9.088519870035481934270409011195, 9.589166291319292280483701314920, 10.73367796985506401003902973602

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