Properties

Label 2-3380-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.33·3-s + 5-s + 0.399·7-s + 2.43·9-s − 1.73·11-s − 2.33·15-s + 0.692·17-s + 5.37·19-s − 0.932·21-s − 0.107·23-s + 25-s + 1.30·27-s − 4.90·29-s − 7.86·31-s + 4.03·33-s + 0.399·35-s − 2.26·37-s + 7.73·41-s + 6.01·43-s + 2.43·45-s − 3.46·47-s − 6.84·49-s − 1.61·51-s + 11.7·53-s − 1.73·55-s − 12.5·57-s − 7.27·59-s + ⋯
L(s)  = 1  − 1.34·3-s + 0.447·5-s + 0.151·7-s + 0.813·9-s − 0.522·11-s − 0.602·15-s + 0.167·17-s + 1.23·19-s − 0.203·21-s − 0.0223·23-s + 0.200·25-s + 0.251·27-s − 0.910·29-s − 1.41·31-s + 0.703·33-s + 0.0675·35-s − 0.372·37-s + 1.20·41-s + 0.916·43-s + 0.363·45-s − 0.505·47-s − 0.977·49-s − 0.226·51-s + 1.60·53-s − 0.233·55-s − 1.65·57-s − 0.947·59-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.069653570\)
\(L(\frac12)\) \(\approx\) \(1.069653570\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.33T + 3T^{2} \)
7 \( 1 - 0.399T + 7T^{2} \)
11 \( 1 + 1.73T + 11T^{2} \)
17 \( 1 - 0.692T + 17T^{2} \)
19 \( 1 - 5.37T + 19T^{2} \)
23 \( 1 + 0.107T + 23T^{2} \)
29 \( 1 + 4.90T + 29T^{2} \)
31 \( 1 + 7.86T + 31T^{2} \)
37 \( 1 + 2.26T + 37T^{2} \)
41 \( 1 - 7.73T + 41T^{2} \)
43 \( 1 - 6.01T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + 7.27T + 59T^{2} \)
61 \( 1 - 8.68T + 61T^{2} \)
67 \( 1 + 1.32T + 67T^{2} \)
71 \( 1 - 3.87T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 + 0.347T + 89T^{2} \)
97 \( 1 - 8.85T + 97T^{2} \)
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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

−8.680372534073023517438310862821, −7.60434053155704866127998371291, −7.12303887980862176512709887175, −6.14186032762986826543061970159, −5.50703376536042793761224632789, −5.17300801012145799866670015733, −4.11925097972989334468279562327, −3.02457914365770236512732633219, −1.82343777503188475784095828983, −0.66275564482904099699357835589, 0.66275564482904099699357835589, 1.82343777503188475784095828983, 3.02457914365770236512732633219, 4.11925097972989334468279562327, 5.17300801012145799866670015733, 5.50703376536042793761224632789, 6.14186032762986826543061970159, 7.12303887980862176512709887175, 7.60434053155704866127998371291, 8.680372534073023517438310862821

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