Properties

Label 2-7350-1.1-c1-0-129
Degree $2$
Conductor $7350$
Sign $-1$
Analytic cond. $58.6900$
Root an. cond. $7.66094$
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 + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s − 2·13-s + 16-s − 2·17-s + 18-s − 2·19-s − 8·23-s + 24-s − 2·26-s + 27-s − 8·29-s − 4·31-s + 32-s − 2·34-s + 36-s + 6·37-s − 2·38-s − 2·39-s − 10·41-s − 2·43-s − 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.458·19-s − 1.66·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s − 1.48·29-s − 0.718·31-s + 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.986·37-s − 0.324·38-s − 0.320·39-s − 1.56·41-s − 0.304·43-s − 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(58.6900\)
Root analytic conductor: \(7.66094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.75972303732766610653670292427, −6.71746052302026106008361856652, −6.23649943255438050233522172382, −5.33714204239010871677222163923, −4.64680835550842315040701992404, −3.88284043167385451077899255203, −3.29818149609701544272153132242, −2.23452117332795942922270569875, −1.76423348275487500936740982432, 0, 1.76423348275487500936740982432, 2.23452117332795942922270569875, 3.29818149609701544272153132242, 3.88284043167385451077899255203, 4.64680835550842315040701992404, 5.33714204239010871677222163923, 6.23649943255438050233522172382, 6.71746052302026106008361856652, 7.75972303732766610653670292427

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