Properties

Label 2-7800-1.1-c1-0-72
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 + 0.476·7-s + 9-s − 3.34·11-s + 13-s + 17-s − 6.77·19-s − 0.476·21-s − 4.77·23-s − 27-s + 7.59·29-s + 7.93·31-s + 3.34·33-s + 7.82·37-s − 39-s + 9.46·41-s + 1.82·43-s − 9.06·47-s − 6.77·49-s − 51-s + 9.50·53-s + 6.77·57-s + 3.16·59-s − 9.59·61-s + 0.476·63-s − 1.34·67-s + 4.77·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.180·7-s + 0.333·9-s − 1.00·11-s + 0.277·13-s + 0.242·17-s − 1.55·19-s − 0.103·21-s − 0.995·23-s − 0.192·27-s + 1.41·29-s + 1.42·31-s + 0.582·33-s + 1.28·37-s − 0.160·39-s + 1.47·41-s + 0.277·43-s − 1.32·47-s − 0.967·49-s − 0.140·51-s + 1.30·53-s + 0.897·57-s + 0.411·59-s − 1.22·61-s + 0.0600·63-s − 0.164·67-s + 0.574·69-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 \( 1 - 0.476T + 7T^{2} \)
11 \( 1 + 3.34T + 11T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 + 6.77T + 19T^{2} \)
23 \( 1 + 4.77T + 23T^{2} \)
29 \( 1 - 7.59T + 29T^{2} \)
31 \( 1 - 7.93T + 31T^{2} \)
37 \( 1 - 7.82T + 37T^{2} \)
41 \( 1 - 9.46T + 41T^{2} \)
43 \( 1 - 1.82T + 43T^{2} \)
47 \( 1 + 9.06T + 47T^{2} \)
53 \( 1 - 9.50T + 53T^{2} \)
59 \( 1 - 3.16T + 59T^{2} \)
61 \( 1 + 9.59T + 61T^{2} \)
67 \( 1 + 1.34T + 67T^{2} \)
71 \( 1 + 4.86T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 + 2.17T + 79T^{2} \)
83 \( 1 - 9.16T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 7.04T + 97T^{2} \)
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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

−7.64534871300336389579749172667, −6.59422264044992996861628876077, −6.17200063863426957627864504018, −5.50779935593322518707527460864, −4.49476337979363732878875839719, −4.30220810689635695705442641791, −2.96639318747627023695066468304, −2.29551530298807110433205226926, −1.14494092282001473116443733280, 0, 1.14494092282001473116443733280, 2.29551530298807110433205226926, 2.96639318747627023695066468304, 4.30220810689635695705442641791, 4.49476337979363732878875839719, 5.50779935593322518707527460864, 6.17200063863426957627864504018, 6.59422264044992996861628876077, 7.64534871300336389579749172667

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