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Properties

Label 2-3680-920.229-c0-0-1
Degree $2$
Conductor $3680$
Sign $1$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·3-s − 5-s − 0.618·7-s + 1.61·9-s − 1.61·11-s − 0.618·13-s + 1.61·15-s − 1.61·17-s + 0.618·19-s + 1.00·21-s − 23-s + 25-s − 27-s − 0.618·31-s + 2.61·33-s + 0.618·35-s + 1.00·39-s − 1.61·41-s − 1.61·45-s − 0.618·49-s + 2.61·51-s + 1.61·55-s − 1.00·57-s + 1.61·61-s − 1.00·63-s + 0.618·65-s + 1.61·69-s + ⋯
L(s)  = 1  − 1.61·3-s − 5-s − 0.618·7-s + 1.61·9-s − 1.61·11-s − 0.618·13-s + 1.61·15-s − 1.61·17-s + 0.618·19-s + 1.00·21-s − 23-s + 25-s − 27-s − 0.618·31-s + 2.61·33-s + 0.618·35-s + 1.00·39-s − 1.61·41-s − 1.61·45-s − 0.618·49-s + 2.61·51-s + 1.61·55-s − 1.00·57-s + 1.61·61-s − 1.00·63-s + 0.618·65-s + 1.61·69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3680\)    =    \(2^{5} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3680} (689, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3680,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1777072238\)
\(L(\frac12)\) \(\approx\) \(0.1777072238\)
\(L(1)\) 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 \( 1 + T \)
23 \( 1 + T \)
good3 \( 1 + 1.61T + T^{2} \)
7 \( 1 + 0.618T + T^{2} \)
11 \( 1 + 1.61T + T^{2} \)
13 \( 1 + 0.618T + T^{2} \)
17 \( 1 + 1.61T + T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 0.618T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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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

−8.557175145039695156976788060268, −7.82693565968870571939511305383, −7.02456307500285172411774986602, −6.59427273069639954116290501285, −5.58399467403044747481781105755, −5.02477271271878968307891602514, −4.36899492878551996592429286406, −3.34340160360658974772073744699, −2.19471880308865485776251519316, −0.36551028701962918915390568527, 0.36551028701962918915390568527, 2.19471880308865485776251519316, 3.34340160360658974772073744699, 4.36899492878551996592429286406, 5.02477271271878968307891602514, 5.58399467403044747481781105755, 6.59427273069639954116290501285, 7.02456307500285172411774986602, 7.82693565968870571939511305383, 8.557175145039695156976788060268

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