Properties

Label 2-7800-1.1-c1-0-106
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 + 9-s + 1.50·11-s + 13-s − 2.72·17-s + 0.726·19-s − 4.72·23-s + 27-s − 7.55·29-s − 3.00·31-s + 1.50·33-s − 5.00·37-s + 39-s + 5.78·41-s + 2.72·43-s − 10.2·47-s − 7·49-s − 2.72·51-s − 7.55·53-s + 0.726·57-s − 12.5·59-s + 6.28·61-s − 12.5·67-s − 4.72·69-s + 4.77·71-s + 12.0·73-s + 5.27·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.333·9-s + 0.453·11-s + 0.277·13-s − 0.661·17-s + 0.166·19-s − 0.985·23-s + 0.192·27-s − 1.40·29-s − 0.540·31-s + 0.261·33-s − 0.823·37-s + 0.160·39-s + 0.903·41-s + 0.415·43-s − 1.49·47-s − 49-s − 0.381·51-s − 1.03·53-s + 0.0962·57-s − 1.62·59-s + 0.804·61-s − 1.53·67-s − 0.569·69-s + 0.567·71-s + 1.40·73-s + 0.593·79-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 \( 1 + 7T^{2} \)
11 \( 1 - 1.50T + 11T^{2} \)
17 \( 1 + 2.72T + 17T^{2} \)
19 \( 1 - 0.726T + 19T^{2} \)
23 \( 1 + 4.72T + 23T^{2} \)
29 \( 1 + 7.55T + 29T^{2} \)
31 \( 1 + 3.00T + 31T^{2} \)
37 \( 1 + 5.00T + 37T^{2} \)
41 \( 1 - 5.78T + 41T^{2} \)
43 \( 1 - 2.72T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + 7.55T + 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 6.28T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 5.27T + 79T^{2} \)
83 \( 1 - 7.78T + 83T^{2} \)
89 \( 1 - 1.78T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−7.72982066223484793036645355814, −6.77803605636580761143779899533, −6.25977116967052907754952054886, −5.40910274308241159375831653547, −4.56001953857520854050563700875, −3.82001045538858944954337429609, −3.22502332833715182511292437417, −2.14738454343177271451854765155, −1.49965898496689832250859761779, 0, 1.49965898496689832250859761779, 2.14738454343177271451854765155, 3.22502332833715182511292437417, 3.82001045538858944954337429609, 4.56001953857520854050563700875, 5.40910274308241159375831653547, 6.25977116967052907754952054886, 6.77803605636580761143779899533, 7.72982066223484793036645355814

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