Properties

Label 2-315-1.1-c1-0-3
Degree $2$
Conductor $315$
Sign $1$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 5-s + 7-s + 3·11-s + 5·13-s + 4·16-s − 3·17-s + 2·19-s − 2·20-s + 6·23-s + 25-s − 2·28-s − 3·29-s − 4·31-s + 35-s + 2·37-s + 12·41-s − 10·43-s − 6·44-s − 9·47-s + 49-s − 10·52-s − 12·53-s + 3·55-s + 8·61-s − 8·64-s + 5·65-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s + 0.377·7-s + 0.904·11-s + 1.38·13-s + 16-s − 0.727·17-s + 0.458·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.377·28-s − 0.557·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s + 1.87·41-s − 1.52·43-s − 0.904·44-s − 1.31·47-s + 1/7·49-s − 1.38·52-s − 1.64·53-s + 0.404·55-s + 1.02·61-s − 64-s + 0.620·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{315} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.273082906\)
\(L(\frac12)\) \(\approx\) \(1.273082906\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
good2 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−11.50769791186352205274303435148, −10.81642134200306905635056067311, −9.502060864672765915422726581101, −9.001872818363131182920485953111, −8.087023964728641191414598204496, −6.69771158952637091109528432278, −5.63914839037632160871019437372, −4.54321060660836499326534703030, −3.45067056060882590557565434961, −1.35738749247200780270220785705, 1.35738749247200780270220785705, 3.45067056060882590557565434961, 4.54321060660836499326534703030, 5.63914839037632160871019437372, 6.69771158952637091109528432278, 8.087023964728641191414598204496, 9.001872818363131182920485953111, 9.502060864672765915422726581101, 10.81642134200306905635056067311, 11.50769791186352205274303435148

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