Properties

Label 2-7800-1.1-c1-0-107
Degree $2$
Conductor $7800$
Sign $-1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2.56·7-s + 9-s − 1.43·11-s − 13-s − 5.68·17-s − 5.12·19-s + 2.56·21-s + 1.43·23-s + 27-s − 2·29-s + 1.12·31-s − 1.43·33-s + 10.8·37-s − 39-s − 9.68·41-s − 6.24·43-s + 1.12·47-s − 0.438·49-s − 5.68·51-s + 0.561·53-s − 5.12·57-s − 8·59-s + 1.68·61-s + 2.56·63-s − 2.24·67-s + 1.43·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.968·7-s + 0.333·9-s − 0.433·11-s − 0.277·13-s − 1.37·17-s − 1.17·19-s + 0.558·21-s + 0.299·23-s + 0.192·27-s − 0.371·29-s + 0.201·31-s − 0.250·33-s + 1.77·37-s − 0.160·39-s − 1.51·41-s − 0.952·43-s + 0.163·47-s − 0.0626·49-s − 0.796·51-s + 0.0771·53-s − 0.678·57-s − 1.04·59-s + 0.215·61-s + 0.322·63-s − 0.274·67-s + 0.173·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(62.2833\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 2.56T + 7T^{2} \)
11 \( 1 + 1.43T + 11T^{2} \)
17 \( 1 + 5.68T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
23 \( 1 - 1.43T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 1.12T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + 9.68T + 41T^{2} \)
43 \( 1 + 6.24T + 43T^{2} \)
47 \( 1 - 1.12T + 47T^{2} \)
53 \( 1 - 0.561T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 1.68T + 61T^{2} \)
67 \( 1 + 2.24T + 67T^{2} \)
71 \( 1 + 7.68T + 71T^{2} \)
73 \( 1 - 0.246T + 73T^{2} \)
79 \( 1 + 8.80T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + 2.31T + 89T^{2} \)
97 \( 1 + 2.80T + 97T^{2} \)
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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

−7.68272922787228454529493909979, −6.84900451905353027028379191787, −6.26615140737730023667786304311, −5.23149284877696839981335537443, −4.58969256206134114059974237496, −4.08567840253363864292127311848, −2.95162915122110165806360903424, −2.23533871389667684011556812253, −1.51581194973951569214074106347, 0, 1.51581194973951569214074106347, 2.23533871389667684011556812253, 2.95162915122110165806360903424, 4.08567840253363864292127311848, 4.58969256206134114059974237496, 5.23149284877696839981335537443, 6.26615140737730023667786304311, 6.84900451905353027028379191787, 7.68272922787228454529493909979

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