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Properties

Label 2-58e2-116.91-c0-0-7
Degree $2$
Conductor $3364$
Sign $0.226 + 0.974i$
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  + (−0.222 + 0.974i)2-s + (0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (0.222 − 0.974i)5-s + (0.222 + 0.974i)6-s + (0.623 − 0.781i)8-s + (0.900 + 0.433i)10-s + (−0.623 − 0.781i)11-s − 12-s + (−0.623 − 0.781i)13-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + (−1.80 − 0.867i)19-s + (−0.623 + 0.781i)20-s + (0.900 − 0.433i)22-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (0.222 − 0.974i)5-s + (0.222 + 0.974i)6-s + (0.623 − 0.781i)8-s + (0.900 + 0.433i)10-s + (−0.623 − 0.781i)11-s − 12-s + (−0.623 − 0.781i)13-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + (−1.80 − 0.867i)19-s + (−0.623 + 0.781i)20-s + (0.900 − 0.433i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.226 + 0.974i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (2759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ 0.226 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.006813369\)
\(L(\frac12)\) \(\approx\) \(1.006813369\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 \)
good3 \( 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.222 + 0.974i)T + (-0.900 - 0.433i)T^{2} \)
7 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.623 + 0.781i)T + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.623 + 0.781i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (1.80 + 0.867i)T + (0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.222 + 0.974i)T + (-0.900 - 0.433i)T^{2} \)
37 \( 1 + (0.222 + 0.974i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-0.222 - 0.974i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (-0.222 + 0.974i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.623 + 0.781i)T^{2} \)
67 \( 1 + (0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.900 - 0.433i)T^{2} \)
79 \( 1 + (0.623 - 0.781i)T + (-0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.900 + 0.433i)T^{2} \)
97 \( 1 + (-0.623 - 0.781i)T^{2} \)
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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

−8.428687617929671183102129859684, −8.142952721218918801820200906506, −7.33985585803053471651547689660, −6.49943973265050884711447765043, −5.57832995753291980706116175467, −5.03931037350675524130346467981, −4.18364821726909193851973271479, −2.96864431766374853449928487463, −1.99686518802196695908015129774, −0.53961023258596781256075265985, 1.91713602232437211148298778452, 2.49376996228624565860460196461, 3.20810098511205916993009093154, 4.14358234399790170967132454573, 4.67819653613373916362819221618, 5.95459113715301455599026625677, 6.89890598195807418164347517294, 7.68166930954514310763484600931, 8.473963169084443690522344099127, 9.057304422119897831853692740124

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