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Properties

Label 2-58e2-116.103-c0-0-2
Degree $2$
Conductor $3364$
Sign $0.976 - 0.214i$
Analytic cond. $1.67885$
Root an. cond. $1.29570$
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  + (−0.433 + 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (−0.900 − 0.433i)5-s + (−0.900 + 0.433i)6-s + (0.974 − 0.222i)8-s + (0.781 − 0.623i)10-s + (−0.974 − 0.222i)11-s − i·12-s + (−0.222 + 0.974i)13-s + (−0.433 − 0.900i)15-s + (−0.222 + 0.974i)16-s + (1.56 − 1.24i)19-s + (0.222 + 0.974i)20-s + (0.623 − 0.781i)22-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (−0.900 − 0.433i)5-s + (−0.900 + 0.433i)6-s + (0.974 − 0.222i)8-s + (0.781 − 0.623i)10-s + (−0.974 − 0.222i)11-s − i·12-s + (−0.222 + 0.974i)13-s + (−0.433 − 0.900i)15-s + (−0.222 + 0.974i)16-s + (1.56 − 1.24i)19-s + (0.222 + 0.974i)20-s + (0.623 − 0.781i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.976 - 0.214i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ 0.976 - 0.214i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8940557990\)
\(L(\frac12)\) \(\approx\) \(0.8940557990\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.433 - 0.900i)T \)
29 \( 1 \)
good3 \( 1 + (-0.781 - 0.623i)T + (0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T + (0.623 + 0.781i)T^{2} \)
7 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.974 + 0.222i)T + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-1.56 + 1.24i)T + (0.222 - 0.974i)T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.433 + 0.900i)T + (-0.623 - 0.781i)T^{2} \)
37 \( 1 + (-0.900 + 0.433i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.433 + 0.900i)T + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.974 - 0.222i)T + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.900 - 0.433i)T + (0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + (0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.623 - 0.781i)T^{2} \)
79 \( 1 + (-0.974 + 0.222i)T + (0.900 - 0.433i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.623 + 0.781i)T^{2} \)
97 \( 1 + (-0.222 + 0.974i)T^{2} \)
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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

−8.731424169568245506125719262010, −8.171894054779018423660136622345, −7.47097630741740478063317784755, −6.88382469841430577389187258455, −5.77024073892172207076642909789, −4.92467019886238128981160336813, −4.32623661752344420497661202116, −3.51033403818665441723994477381, −2.39958320209272026728916814160, −0.64306720127108594892694452149, 1.19353745133900104025528796165, 2.36785801434197633485144050864, 3.11423880948684136085213344607, 3.57123416015577754396135018413, 4.80619732692651228734648575281, 5.57684236708911929780549839133, 7.09120504952310950700954212200, 7.69826458827011952379970151495, 7.914807233922637991800229344701, 8.576996886135388406467929984562

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