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Properties

Label 2-3680-115.79-c0-0-2
Degree $2$
Conductor $3680$
Sign $0.689 - 0.724i$
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  + (−0.0771 + 0.120i)3-s + (0.909 + 0.415i)5-s + (0.919 + 0.270i)7-s + (0.406 + 0.891i)9-s + (−0.120 + 0.0771i)15-s + (−0.103 + 0.0895i)21-s + (0.0713 − 0.997i)23-s + (0.654 + 0.755i)25-s + (−0.279 − 0.0401i)27-s + (0.118 + 0.822i)29-s + (0.724 + 0.627i)35-s + (0.234 − 0.512i)41-s + (−1.34 − 0.865i)43-s + 0.979i·45-s + 0.698i·47-s + ⋯
L(s)  = 1  + (−0.0771 + 0.120i)3-s + (0.909 + 0.415i)5-s + (0.919 + 0.270i)7-s + (0.406 + 0.891i)9-s + (−0.120 + 0.0771i)15-s + (−0.103 + 0.0895i)21-s + (0.0713 − 0.997i)23-s + (0.654 + 0.755i)25-s + (−0.279 − 0.0401i)27-s + (0.118 + 0.822i)29-s + (0.724 + 0.627i)35-s + (0.234 − 0.512i)41-s + (−1.34 − 0.865i)43-s + 0.979i·45-s + 0.698i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3680\)    =    \(2^{5} \cdot 5 \cdot 23\)
Sign: $0.689 - 0.724i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3680} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3680,\ (\ :0),\ 0.689 - 0.724i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.706560019\)
\(L(\frac12)\) \(\approx\) \(1.706560019\)
\(L(1)\) 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 \( 1 + (-0.909 - 0.415i)T \)
23 \( 1 + (-0.0713 + 0.997i)T \)
good3 \( 1 + (0.0771 - 0.120i)T + (-0.415 - 0.909i)T^{2} \)
7 \( 1 + (-0.919 - 0.270i)T + (0.841 + 0.540i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (-0.841 + 0.540i)T^{2} \)
17 \( 1 + (-0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \)
31 \( 1 + (0.415 - 0.909i)T^{2} \)
37 \( 1 + (-0.654 + 0.755i)T^{2} \)
41 \( 1 + (-0.234 + 0.512i)T + (-0.654 - 0.755i)T^{2} \)
43 \( 1 + (1.34 + 0.865i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 - 0.698iT - T^{2} \)
53 \( 1 + (0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (0.153 + 0.239i)T + (-0.415 + 0.909i)T^{2} \)
67 \( 1 + (1.27 + 1.47i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.176 - 0.386i)T + (-0.654 + 0.755i)T^{2} \)
89 \( 1 + (0.817 - 1.27i)T + (-0.415 - 0.909i)T^{2} \)
97 \( 1 + (-0.654 - 0.755i)T^{2} \)
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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

−8.740610964730217688735745902124, −8.136479901490347918001946858862, −7.28472162959983889751468680732, −6.61741376527488278240577584958, −5.70219908660546511790689061725, −5.04869667421975489775204988808, −4.46049989820495336772402456251, −3.18344977007774200393772586346, −2.20366278148597425901791797733, −1.55768266243257061166406785378, 1.13710186858177380833016591470, 1.84123293994853282650201560220, 3.05455080388466121673838357164, 4.14202856561945714309365101517, 4.83489120149647690133906922251, 5.65697336306966143261789718420, 6.32458505954962471000776436831, 7.11585262061027148946908064395, 7.912308749042232027070168257787, 8.637380599253414297056626762820

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