Properties

Label 2-3150-1.1-c1-0-26
Degree $2$
Conductor $3150$
Sign $1$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
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-s + 4-s + 7-s + 8-s + 5·11-s + 6·13-s + 14-s + 16-s − 17-s − 3·19-s + 5·22-s + 6·26-s + 28-s + 6·29-s − 4·31-s + 32-s − 34-s − 8·37-s − 3·38-s − 11·41-s + 8·43-s + 5·44-s + 2·47-s + 49-s + 6·52-s + 4·53-s + 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.50·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.688·19-s + 1.06·22-s + 1.17·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 1.31·37-s − 0.486·38-s − 1.71·41-s + 1.21·43-s + 0.753·44-s + 0.291·47-s + 1/7·49-s + 0.832·52-s + 0.549·53-s + 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.683106985\)
\(L(\frac12)\) \(\approx\) \(3.683106985\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.745835108946327868584466781878, −7.967535843941389291937593485802, −6.82833712853147272597306626000, −6.46177038230604663817255689879, −5.68728163064030395991011176054, −4.71485812600927611791203093272, −3.91736185403180141186201303911, −3.39930630284680276801854080586, −2.01283634789248302897310187156, −1.16422993590034766917798668900, 1.16422993590034766917798668900, 2.01283634789248302897310187156, 3.39930630284680276801854080586, 3.91736185403180141186201303911, 4.71485812600927611791203093272, 5.68728163064030395991011176054, 6.46177038230604663817255689879, 6.82833712853147272597306626000, 7.967535843941389291937593485802, 8.745835108946327868584466781878

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