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Properties

Label 2-371-371.307-c0-0-0
Degree $2$
Conductor $371$
Sign $-0.990 + 0.140i$
Analytic cond. $0.185153$
Root an. cond. $0.430294$
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  + (−0.627 + 1.65i)2-s + (−1.59 − 1.41i)4-s + (−0.354 + 0.935i)7-s + (1.78 − 0.935i)8-s + (−0.970 + 0.239i)9-s + (−1.10 + 1.59i)11-s + (−1.32 − 1.17i)14-s + (0.171 + 1.41i)16-s + (0.213 − 1.75i)18-s + (−1.95 − 2.83i)22-s + 1.13·23-s + (0.885 − 0.464i)25-s + (1.89 − 0.992i)28-s + (1.00 + 1.45i)29-s + (−0.500 − 0.123i)32-s + ⋯
L(s)  = 1  + (−0.627 + 1.65i)2-s + (−1.59 − 1.41i)4-s + (−0.354 + 0.935i)7-s + (1.78 − 0.935i)8-s + (−0.970 + 0.239i)9-s + (−1.10 + 1.59i)11-s + (−1.32 − 1.17i)14-s + (0.171 + 1.41i)16-s + (0.213 − 1.75i)18-s + (−1.95 − 2.83i)22-s + 1.13·23-s + (0.885 − 0.464i)25-s + (1.89 − 0.992i)28-s + (1.00 + 1.45i)29-s + (−0.500 − 0.123i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(371\)    =    \(7 \cdot 53\)
Sign: $-0.990 + 0.140i$
Analytic conductor: \(0.185153\)
Root analytic conductor: \(0.430294\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{371} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 371,\ (\ :0),\ -0.990 + 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4266875022\)
\(L(\frac12)\) \(\approx\) \(0.4266875022\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.354 - 0.935i)T \)
53 \( 1 + (0.970 + 0.239i)T \)
good2 \( 1 + (0.627 - 1.65i)T + (-0.748 - 0.663i)T^{2} \)
3 \( 1 + (0.970 - 0.239i)T^{2} \)
5 \( 1 + (-0.885 + 0.464i)T^{2} \)
11 \( 1 + (1.10 - 1.59i)T + (-0.354 - 0.935i)T^{2} \)
13 \( 1 + (-0.120 + 0.992i)T^{2} \)
17 \( 1 + (-0.568 - 0.822i)T^{2} \)
19 \( 1 + (-0.120 + 0.992i)T^{2} \)
23 \( 1 - 1.13T + T^{2} \)
29 \( 1 + (-1.00 - 1.45i)T + (-0.354 + 0.935i)T^{2} \)
31 \( 1 + (0.354 - 0.935i)T^{2} \)
37 \( 1 + (0.180 + 1.48i)T + (-0.970 + 0.239i)T^{2} \)
41 \( 1 + (0.354 + 0.935i)T^{2} \)
43 \( 1 + (-0.0290 + 0.239i)T + (-0.970 - 0.239i)T^{2} \)
47 \( 1 + (-0.885 - 0.464i)T^{2} \)
59 \( 1 + (-0.885 - 0.464i)T^{2} \)
61 \( 1 + (-0.568 + 0.822i)T^{2} \)
67 \( 1 + (-1.45 - 1.28i)T + (0.120 + 0.992i)T^{2} \)
71 \( 1 + (0.0854 - 0.704i)T + (-0.970 - 0.239i)T^{2} \)
73 \( 1 + (-0.568 - 0.822i)T^{2} \)
79 \( 1 + (-0.251 - 0.663i)T + (-0.748 + 0.663i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.568 - 0.822i)T^{2} \)
97 \( 1 + (-0.885 + 0.464i)T^{2} \)
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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

−12.33983632759602061247609940857, −10.85645552578273552942678745012, −9.890814209882057775218630171823, −8.979227869620791858424977929456, −8.387415089456495092883982689937, −7.34811381439462284507029245400, −6.59358398171604727303406723627, −5.38857651049702912652012165047, −4.94295778836324010024975949336, −2.63849409013228794111383521913, 0.74954375146009992019410616815, 2.85815344634763632123239861641, 3.35483660965339845819918442353, 4.86413799125797615088321870398, 6.32448631224053300505594292398, 7.940995706403501094260244245854, 8.567627267376057977331659505610, 9.523271500000378081739687793516, 10.51109109226188881373680574650, 11.01047008427115214062760847760

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