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Properties

Label 2-3332-3332.2923-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.138 + 0.990i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (−0.0747 − 0.997i)2-s + (0.563 + 0.173i)3-s + (−0.988 + 0.149i)4-s + (0.131 − 0.574i)6-s + (0.974 − 0.222i)7-s + (0.222 + 0.974i)8-s + (−0.539 − 0.367i)9-s + (1.64 − 1.12i)11-s + (−0.582 − 0.0878i)12-s + (0.134 + 0.0648i)13-s + (−0.294 − 0.955i)14-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−0.326 + 0.565i)18-s + (0.587 + 0.0440i)21-s + (−1.24 − 1.55i)22-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (0.563 + 0.173i)3-s + (−0.988 + 0.149i)4-s + (0.131 − 0.574i)6-s + (0.974 − 0.222i)7-s + (0.222 + 0.974i)8-s + (−0.539 − 0.367i)9-s + (1.64 − 1.12i)11-s + (−0.582 − 0.0878i)12-s + (0.134 + 0.0648i)13-s + (−0.294 − 0.955i)14-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−0.326 + 0.565i)18-s + (0.587 + 0.0440i)21-s + (−1.24 − 1.55i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.138 + 0.990i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2923, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.559813590\)
\(L(\frac12)\) \(\approx\) \(1.559813590\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0747 + 0.997i)T \)
7 \( 1 + (-0.974 + 0.222i)T \)
17 \( 1 + (-0.365 - 0.930i)T \)
good3 \( 1 + (-0.563 - 0.173i)T + (0.826 + 0.563i)T^{2} \)
5 \( 1 + (-0.0747 + 0.997i)T^{2} \)
11 \( 1 + (-1.64 + 1.12i)T + (0.365 - 0.930i)T^{2} \)
13 \( 1 + (-0.134 - 0.0648i)T + (0.623 + 0.781i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.317 - 0.807i)T + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.974 - 1.68i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.988 - 0.149i)T^{2} \)
53 \( 1 + (-1.44 + 0.218i)T + (0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (-0.955 - 0.294i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.848 + 1.06i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (0.563 + 0.975i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (1.57 + 1.07i)T + (0.365 + 0.930i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.619116031453930910147080384323, −8.441487604820978061517895637737, −7.40167221607972367614817489777, −6.20051170072508353142062040719, −5.54198496784861029943458864762, −4.41603966073299315516030353810, −3.70035151841051803412871416365, −3.24718549733350613309028212427, −1.93159265085208418909032408416, −1.11637839363043069283824711332, 1.36233289576536783533925602750, 2.39489153686810657834653619247, 3.77851012184916587119019539336, 4.42513214702375879433230474366, 5.29855395690687887493890130888, 5.94694402924732810504768892924, 7.06587385162726456630109650342, 7.38893791859686156573699503159, 8.191649188750002291752937344072, 8.879228673527052182604219337966

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