Properties

Label 2-58e2-116.7-c0-0-2
Degree $2$
Conductor $3364$
Sign $0.692 - 0.721i$
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)4-s + (0.400 + 1.75i)5-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (1.62 − 0.781i)10-s + (−0.277 + 0.347i)13-s + (0.623 − 0.781i)16-s + 1.24·17-s + (0.623 − 0.781i)18-s + (−1.12 − 1.40i)20-s + (−2.02 + 0.974i)25-s + (0.400 + 0.193i)26-s + (−0.900 − 0.433i)32-s + (−0.277 − 1.21i)34-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.400 + 1.75i)5-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (1.62 − 0.781i)10-s + (−0.277 + 0.347i)13-s + (0.623 − 0.781i)16-s + 1.24·17-s + (0.623 − 0.781i)18-s + (−1.12 − 1.40i)20-s + (−2.02 + 0.974i)25-s + (0.400 + 0.193i)26-s + (−0.900 − 0.433i)32-s + (−0.277 − 1.21i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.692 - 0.721i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (2327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ 0.692 - 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.121203848\)
\(L(\frac12)\) \(\approx\) \(1.121203848\)
\(L(1)\) 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 \( 1 + (-0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
7 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 - 1.24T + T^{2} \)
19 \( 1 + (-0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.900 - 0.433i)T^{2} \)
37 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
41 \( 1 + 0.445T + T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 + (0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
79 \( 1 + (0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
97 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
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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

−9.175403364784237571179910303186, −8.050255486303782597066166220230, −7.48672099894716418872958178484, −6.85214953102018346238944407294, −5.83018702091182914527600056238, −4.97991389072583732029196506121, −3.92946201051589895232279755522, −3.17911458557812779090465778757, −2.41156502618669270674940815144, −1.61333100877314964364469782377, 0.791797593988843908331206252063, 1.59795866858031388158649661879, 3.46053538549831930717725138565, 4.40068654994892696309010876951, 5.00214422105887108544579962481, 5.70706464801027237326288501363, 6.31715819848333832374393473624, 7.41746935130896715609711495067, 7.897850736487838868975780620394, 8.877769435419334062050107220299

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