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Properties

Label 2-58e2-116.115-c0-0-5
Degree $2$
Conductor $3364$
Sign $0.648 + 0.760i$
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  + i·2-s − 4-s + 0.445·5-s − i·8-s − 9-s + 0.445i·10-s − 1.24·13-s + 16-s − 1.80i·17-s − i·18-s − 0.445·20-s − 0.801·25-s − 1.24i·26-s + i·32-s + 1.80·34-s + ⋯
L(s)  = 1  + i·2-s − 4-s + 0.445·5-s − i·8-s − 9-s + 0.445i·10-s − 1.24·13-s + 16-s − 1.80i·17-s − i·18-s − 0.445·20-s − 0.801·25-s − 1.24i·26-s + i·32-s + 1.80·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.648 + 0.760i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (3363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ 0.648 + 0.760i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5876358245\)
\(L(\frac12)\) \(\approx\) \(0.5876358245\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
29 \( 1 \)
good3 \( 1 + T^{2} \)
5 \( 1 - 0.445T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 1.24T + T^{2} \)
17 \( 1 + 1.80iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 1.24iT - T^{2} \)
41 \( 1 + 1.24iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.80T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.80iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.80iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 0.445iT - T^{2} \)
97 \( 1 + 1.24iT - T^{2} \)
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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

−8.670102803698938680463092920523, −7.76898134290798537363372501856, −7.26132467519314249446172685842, −6.49920220690350589931651990536, −5.52418540534631227720228160719, −5.24575053270127044566464746303, −4.30021891481328439543972360044, −3.15627239374601853934288589326, −2.24052713000643507955345653223, −0.33681256382783303418447606707, 1.51262253345647225706528854608, 2.41533019054601121800182990252, 3.18354641384164844146912926993, 4.16046473933419011488277322498, 4.98982208042376053664321506554, 5.77822785408932160657714715988, 6.42178090219155471905114325494, 7.78746348983144823448197675525, 8.232249558105802579087656193655, 9.084386415355007389999585947578

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