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Properties

Label 2-58e2-116.23-c0-0-4
Degree $2$
Conductor $3364$
Sign $0.831 + 0.556i$
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  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.777 − 0.974i)5-s + (0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.277 + 1.21i)10-s + (1.62 − 0.781i)13-s + (−0.900 + 0.433i)16-s + 0.445·17-s + (0.900 − 0.433i)18-s + (−1.12 − 0.541i)20-s + (−0.123 − 0.541i)25-s + (−0.400 + 1.75i)26-s + (0.222 − 0.974i)32-s + (−0.277 + 0.347i)34-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.777 − 0.974i)5-s + (0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.277 + 1.21i)10-s + (1.62 − 0.781i)13-s + (−0.900 + 0.433i)16-s + 0.445·17-s + (0.900 − 0.433i)18-s + (−1.12 − 0.541i)20-s + (−0.123 − 0.541i)25-s + (−0.400 + 1.75i)26-s + (0.222 − 0.974i)32-s + (−0.277 + 0.347i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.831 + 0.556i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (1415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ 0.831 + 0.556i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.015487854\)
\(L(\frac12)\) \(\approx\) \(1.015487854\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 \)
good3 \( 1 + (0.900 + 0.433i)T^{2} \)
5 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
7 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 - 0.445T + T^{2} \)
19 \( 1 + (0.900 - 0.433i)T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.222 + 0.974i)T^{2} \)
37 \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \)
41 \( 1 - 1.80T + T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 + (-0.623 - 0.781i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \)
97 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
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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

−8.476707929783674174576933605884, −8.416478013576816841826471822485, −7.30255178581485750172291700519, −6.32696320375098503659385464771, −5.63796927757336251394501851829, −5.47256015135556044840579402525, −4.26965298941069433955195191218, −3.15916597468984618352747658117, −1.73823405018307846689934745110, −0.818419328000910512249092922626, 1.42184748827165518370066426708, 2.32373093776032917349730889975, 3.13697159374233282198510069712, 3.83876762393694147539024673985, 5.02913384856094874763505671384, 6.09321127310693076287190573944, 6.56647354358952750431580695945, 7.55355800151653603723227669683, 8.331803200771207131535967677069, 8.937202321785261483672984715651

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