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Properties

Label 2-3680-115.74-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9652875344\)
\(L(\frac12)\) \(\approx\) \(0.9652875344\)
\(L(1)\) 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

−8.560193974750008602469099279060, −8.075325389422472549580709262335, −7.11292524519667267951582690642, −6.26089357353762796235999181071, −5.95628727530171735000203530191, −5.26624891031205343491754902225, −4.74389897609269384280066821743, −2.87561263759380585427731362651, −1.98714900030025530826807517106, −1.44894749247882164229499877326, 0.68702035729178542758674349644, 1.85776973504346510548650497415, 3.54074337422279746376624939766, 4.22985984311349061154390767971, 4.90447337509153944189343570625, 5.42017877899697221556293168490, 6.34661731010257753153128378253, 6.86974026462421421873504258963, 7.84990408288238963638085731318, 8.816338358369167226114289233400

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