Properties

Label 2-3150-1.1-c1-0-39
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 − 4·13-s − 14-s + 16-s − 2·17-s − 8·19-s + 8·23-s − 4·26-s − 28-s − 8·29-s + 4·31-s + 32-s − 2·34-s + 8·37-s − 8·38-s − 12·41-s − 8·43-s + 8·46-s − 4·47-s + 49-s − 4·52-s − 6·53-s − 56-s − 8·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 1.83·19-s + 1.66·23-s − 0.784·26-s − 0.188·28-s − 1.48·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 1.31·37-s − 1.29·38-s − 1.87·41-s − 1.21·43-s + 1.17·46-s − 0.583·47-s + 1/7·49-s − 0.554·52-s − 0.824·53-s − 0.133·56-s − 1.05·58-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: Trivial
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 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−8.245799437086343760393736146379, −7.37809344928501113201345267535, −6.66222910486457306007719363045, −6.13948866156581164180828120483, −4.98097563320052319862799855392, −4.60504583712161652651498845302, −3.52477719735270810583168107695, −2.69330686345084354832872734018, −1.77998975598385783353163081350, 0, 1.77998975598385783353163081350, 2.69330686345084354832872734018, 3.52477719735270810583168107695, 4.60504583712161652651498845302, 4.98097563320052319862799855392, 6.13948866156581164180828120483, 6.66222910486457306007719363045, 7.37809344928501113201345267535, 8.245799437086343760393736146379

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