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Properties

Label 2-3680-115.74-c0-0-3
Degree $2$
Conductor $3680$
Sign $0.789 - 0.613i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
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  + (1.32 + 1.14i)3-s + (0.989 + 0.142i)5-s + (−0.778 − 1.70i)7-s + (0.296 + 2.06i)9-s + (1.14 + 1.32i)15-s + (0.926 − 3.15i)21-s + (0.877 − 0.479i)23-s + (0.959 + 0.281i)25-s + (−1.02 + 1.59i)27-s + (0.239 − 0.153i)29-s + (−0.527 − 1.79i)35-s + (−0.258 + 1.80i)41-s + (1.27 − 1.47i)43-s + 2.08i·45-s + 1.60i·47-s + ⋯
L(s)  = 1  + (1.32 + 1.14i)3-s + (0.989 + 0.142i)5-s + (−0.778 − 1.70i)7-s + (0.296 + 2.06i)9-s + (1.14 + 1.32i)15-s + (0.926 − 3.15i)21-s + (0.877 − 0.479i)23-s + (0.959 + 0.281i)25-s + (−1.02 + 1.59i)27-s + (0.239 − 0.153i)29-s + (−0.527 − 1.79i)35-s + (−0.258 + 1.80i)41-s + (1.27 − 1.47i)43-s + 2.08i·45-s + 1.60i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3680\)    =    \(2^{5} \cdot 5 \cdot 23\)
Sign: $0.789 - 0.613i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3680} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3680,\ (\ :0),\ 0.789 - 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.267211207\)
\(L(\frac12)\) \(\approx\) \(2.267211207\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.989 - 0.142i)T \)
23 \( 1 + (-0.877 + 0.479i)T \)
good3 \( 1 + (-1.32 - 1.14i)T + (0.142 + 0.989i)T^{2} \)
7 \( 1 + (0.778 + 1.70i)T + (-0.654 + 0.755i)T^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (0.654 + 0.755i)T^{2} \)
17 \( 1 + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (-0.959 + 0.281i)T^{2} \)
41 \( 1 + (0.258 - 1.80i)T + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (-1.27 + 1.47i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 - 1.60iT - T^{2} \)
53 \( 1 + (-0.654 + 0.755i)T^{2} \)
59 \( 1 + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (1.27 - 1.10i)T + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (1.91 + 0.562i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.654 + 0.755i)T^{2} \)
83 \( 1 + (-0.0203 - 0.141i)T + (-0.959 + 0.281i)T^{2} \)
89 \( 1 + (0.425 + 0.368i)T + (0.142 + 0.989i)T^{2} \)
97 \( 1 + (-0.959 - 0.281i)T^{2} \)
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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

−9.089548510878726854365172049263, −8.147640635871509299548792725020, −7.37685016509674651459990320784, −6.71380189088155510385497566992, −5.76973081885602705804559023641, −4.55312312818122124156004550135, −4.25516926307492807537639409537, −3.13110698383590458763648219259, −2.81817002232678690283055065624, −1.39908899018111617856972114180, 1.40240827074021502317692661348, 2.31008480161320908502356862418, 2.74510757571240132677388845825, 3.54017542182014527592450606937, 5.08602799505949565369705147459, 5.86224251056953761648243042505, 6.45454781063862850822391209456, 7.13594180499258808687952051162, 8.016261259696828220422260130270, 8.897692638595096564786397696896

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