Properties

Label 2-3380-1.1-c1-0-19
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  − 2.52·3-s − 5-s + 4.54·7-s + 3.36·9-s + 2.52·11-s + 2.52·15-s + 5.14·17-s + 7.52·19-s − 11.4·21-s + 2.63·23-s + 25-s − 0.933·27-s − 8.61·29-s + 8.44·31-s − 6.36·33-s − 4.54·35-s + 6.98·37-s + 2.66·41-s − 9.74·43-s − 3.36·45-s − 8.15·47-s + 13.6·49-s − 12.9·51-s − 1.43·53-s − 2.52·55-s − 18.9·57-s + 0.0833·59-s + ⋯
L(s)  = 1  − 1.45·3-s − 0.447·5-s + 1.71·7-s + 1.12·9-s + 0.760·11-s + 0.651·15-s + 1.24·17-s + 1.72·19-s − 2.50·21-s + 0.549·23-s + 0.200·25-s − 0.179·27-s − 1.60·29-s + 1.51·31-s − 1.10·33-s − 0.768·35-s + 1.14·37-s + 0.415·41-s − 1.48·43-s − 0.502·45-s − 1.18·47-s + 1.95·49-s − 1.81·51-s − 0.197·53-s − 0.340·55-s − 2.51·57-s + 0.0108·59-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: \(0\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.526975704\)
\(L(\frac12)\) \(\approx\) \(1.526975704\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.52T + 3T^{2} \)
7 \( 1 - 4.54T + 7T^{2} \)
11 \( 1 - 2.52T + 11T^{2} \)
17 \( 1 - 5.14T + 17T^{2} \)
19 \( 1 - 7.52T + 19T^{2} \)
23 \( 1 - 2.63T + 23T^{2} \)
29 \( 1 + 8.61T + 29T^{2} \)
31 \( 1 - 8.44T + 31T^{2} \)
37 \( 1 - 6.98T + 37T^{2} \)
41 \( 1 - 2.66T + 41T^{2} \)
43 \( 1 + 9.74T + 43T^{2} \)
47 \( 1 + 8.15T + 47T^{2} \)
53 \( 1 + 1.43T + 53T^{2} \)
59 \( 1 - 0.0833T + 59T^{2} \)
61 \( 1 + 0.818T + 61T^{2} \)
67 \( 1 - 5.08T + 67T^{2} \)
71 \( 1 + 7.78T + 71T^{2} \)
73 \( 1 - 0.148T + 73T^{2} \)
79 \( 1 - 7.18T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 - 3.05T + 89T^{2} \)
97 \( 1 - 8.27T + 97T^{2} \)
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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

−8.412209472760972986591275262767, −7.74206784735507007445595307772, −7.19888956322743745786999257025, −6.22642763597963932112853180916, −5.40717738314810979481844148230, −5.00106490954386773112740679258, −4.24289667497706825188472466515, −3.17304319054807237154532330479, −1.51271341052675582796799305511, −0.911472599080910970591483310459, 0.911472599080910970591483310459, 1.51271341052675582796799305511, 3.17304319054807237154532330479, 4.24289667497706825188472466515, 5.00106490954386773112740679258, 5.40717738314810979481844148230, 6.22642763597963932112853180916, 7.19888956322743745786999257025, 7.74206784735507007445595307772, 8.412209472760972986591275262767

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