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Properties

Label 2-3380-260.179-c0-0-13
Degree $2$
Conductor $3380$
Sign $-0.0502 + 0.998i$
Analytic cond. $1.68683$
Root an. cond. $1.29878$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.623 + 1.07i)3-s + (0.499 − 0.866i)4-s − i·5-s + (1.07 + 0.623i)6-s + (−1.56 − 0.900i)7-s − 0.999i·8-s + (−0.277 + 0.480i)9-s + (−0.5 − 0.866i)10-s + 1.24·12-s − 1.80·14-s + (1.07 − 0.623i)15-s + (−0.5 − 0.866i)16-s + 0.554i·18-s + (−0.866 − 0.499i)20-s − 2.24i·21-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.623 + 1.07i)3-s + (0.499 − 0.866i)4-s − i·5-s + (1.07 + 0.623i)6-s + (−1.56 − 0.900i)7-s − 0.999i·8-s + (−0.277 + 0.480i)9-s + (−0.5 − 0.866i)10-s + 1.24·12-s − 1.80·14-s + (1.07 − 0.623i)15-s + (−0.5 − 0.866i)16-s + 0.554i·18-s + (−0.866 − 0.499i)20-s − 2.24i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0502 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0502 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.0502 + 0.998i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -0.0502 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.060122266\)
\(L(\frac12)\) \(\approx\) \(2.060122266\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + iT \)
13 \( 1 \)
good3 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (1.56 + 0.900i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.56 + 0.900i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - 0.445iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.24iT - T^{2} \)
89 \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.081895342275751436479535881560, −7.87347227691849934111732361990, −6.97486137313162398978862986290, −6.12317866578448815587374717146, −5.36496721485964395945794058848, −4.38873640725518369716860938170, −3.92477737387531372304437657685, −3.38035240969221438299654970410, −2.37819317041807503626218672985, −0.817166871027631114444307414590, 1.98966659284296221469051126806, 2.89939325284227409844063259616, 3.07143685917097775504318720981, 4.20603142907028183589621921382, 5.58920693459498926954262760951, 6.10826047141591979196388284879, 6.87387664586204859894028121568, 7.12847204189656003154473898022, 7.996829532980341195621254162113, 8.772184911705222712704307267184

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