Properties

Label 2-3380-1.1-c1-0-44
Degree $2$
Conductor $3380$
Sign $-1$
Analytic cond. $26.9894$
Root an. cond. $5.19513$
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  + 1.24·3-s + 5-s − 1.24·7-s − 1.44·9-s − 0.198·11-s + 1.24·15-s − 4.85·17-s − 1.35·19-s − 1.55·21-s − 1.44·23-s + 25-s − 5.54·27-s + 1.13·29-s + 6.85·31-s − 0.246·33-s − 1.24·35-s − 3·37-s + 3.54·41-s − 7.89·43-s − 1.44·45-s − 8.87·47-s − 5.44·49-s − 6.04·51-s − 1.86·53-s − 0.198·55-s − 1.69·57-s − 0.878·59-s + ⋯
L(s)  = 1  + 0.719·3-s + 0.447·5-s − 0.471·7-s − 0.481·9-s − 0.0597·11-s + 0.321·15-s − 1.17·17-s − 0.311·19-s − 0.339·21-s − 0.301·23-s + 0.200·25-s − 1.06·27-s + 0.211·29-s + 1.23·31-s − 0.0429·33-s − 0.210·35-s − 0.493·37-s + 0.553·41-s − 1.20·43-s − 0.215·45-s − 1.29·47-s − 0.777·49-s − 0.847·51-s − 0.256·53-s − 0.0267·55-s − 0.224·57-s − 0.114·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(26.9894\)
Root analytic conductor: \(5.19513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
13 \( 1 \)
good3 \( 1 - 1.24T + 3T^{2} \)
7 \( 1 + 1.24T + 7T^{2} \)
11 \( 1 + 0.198T + 11T^{2} \)
17 \( 1 + 4.85T + 17T^{2} \)
19 \( 1 + 1.35T + 19T^{2} \)
23 \( 1 + 1.44T + 23T^{2} \)
29 \( 1 - 1.13T + 29T^{2} \)
31 \( 1 - 6.85T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 - 3.54T + 41T^{2} \)
43 \( 1 + 7.89T + 43T^{2} \)
47 \( 1 + 8.87T + 47T^{2} \)
53 \( 1 + 1.86T + 53T^{2} \)
59 \( 1 + 0.878T + 59T^{2} \)
61 \( 1 + 8.19T + 61T^{2} \)
67 \( 1 - 0.207T + 67T^{2} \)
71 \( 1 - 0.664T + 71T^{2} \)
73 \( 1 + 1.72T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + 1.31T + 97T^{2} \)
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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

−8.428334832582353120145824239496, −7.62942552666571055108448276727, −6.57313247131949902715385825857, −6.21660791780946572705888083799, −5.14889997614206993723859099882, −4.32013231961836052914529360586, −3.28278666410211767091212967150, −2.62658691245796937738339615495, −1.71552400392558046669511645146, 0, 1.71552400392558046669511645146, 2.62658691245796937738339615495, 3.28278666410211767091212967150, 4.32013231961836052914529360586, 5.14889997614206993723859099882, 6.21660791780946572705888083799, 6.57313247131949902715385825857, 7.62942552666571055108448276727, 8.428334832582353120145824239496

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