Properties

Label 2-3150-1.1-c1-0-46
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 2·11-s − 6·13-s + 14-s + 16-s + 4·17-s − 6·19-s − 2·22-s − 8·23-s − 6·26-s + 28-s − 6·29-s − 2·31-s + 32-s + 4·34-s − 4·37-s − 6·38-s − 2·41-s − 4·43-s − 2·44-s − 8·46-s + 8·47-s + 49-s − 6·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.603·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 1.37·19-s − 0.426·22-s − 1.66·23-s − 1.17·26-s + 0.188·28-s − 1.11·29-s − 0.359·31-s + 0.176·32-s + 0.685·34-s − 0.657·37-s − 0.973·38-s − 0.312·41-s − 0.609·43-s − 0.301·44-s − 1.17·46-s + 1.16·47-s + 1/7·49-s − 0.832·52-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: \(1\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−8.036835570616111132687795568095, −7.57781723598090862974782182719, −6.81080374028761293998094873350, −5.78793574564528183106350649884, −5.27167215176390503105650672282, −4.43244936590476019100850584085, −3.67274830953119673001999303428, −2.51052952009881589815450760658, −1.88342392731401272278008367686, 0, 1.88342392731401272278008367686, 2.51052952009881589815450760658, 3.67274830953119673001999303428, 4.43244936590476019100850584085, 5.27167215176390503105650672282, 5.78793574564528183106350649884, 6.81080374028761293998094873350, 7.57781723598090862974782182719, 8.036835570616111132687795568095

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