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Properties

Label 2-385-7.4-c1-0-9
Degree $2$
Conductor $385$
Sign $-0.865 - 0.501i$
Analytic cond. $3.07424$
Root an. cond. $1.75335$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.28 + 2.23i)2-s + (0.257 − 0.445i)3-s + (−2.31 + 4.01i)4-s + (−0.5 − 0.866i)5-s + 1.32·6-s + (1.46 + 2.20i)7-s − 6.79·8-s + (1.36 + 2.36i)9-s + (1.28 − 2.23i)10-s + (−0.5 + 0.866i)11-s + (1.19 + 2.06i)12-s − 1.97·13-s + (−3.03 + 6.10i)14-s − 0.514·15-s + (−4.11 − 7.13i)16-s + (1.19 − 2.06i)17-s + ⋯
L(s)  = 1  + (0.910 + 1.57i)2-s + (0.148 − 0.257i)3-s + (−1.15 + 2.00i)4-s + (−0.223 − 0.387i)5-s + 0.541·6-s + (0.552 + 0.833i)7-s − 2.40·8-s + (0.455 + 0.789i)9-s + (0.407 − 0.705i)10-s + (−0.150 + 0.261i)11-s + (0.344 + 0.597i)12-s − 0.546·13-s + (−0.811 + 1.63i)14-s − 0.132·15-s + (−1.02 − 1.78i)16-s + (0.289 − 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.865 - 0.501i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ -0.865 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.535776 + 1.99273i\)
\(L(\frac12)\) \(\approx\) \(0.535776 + 1.99273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.46 - 2.20i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.28 - 2.23i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.257 + 0.445i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 + 1.97T + 13T^{2} \)
17 \( 1 + (-1.19 + 2.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.27 + 3.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.35 - 2.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.93T + 29T^{2} \)
31 \( 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.06 + 5.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.20T + 41T^{2} \)
43 \( 1 - 9.15T + 43T^{2} \)
47 \( 1 + (-3.72 - 6.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.14 + 10.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.929 - 1.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.466 - 0.807i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.77 + 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + (-3.58 + 6.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.410 - 0.710i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.83T + 83T^{2} \)
89 \( 1 + (5.29 + 9.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.02T + 97T^{2} \)
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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

−12.22930277699832303924009033676, −11.05027613203315151182419965887, −9.412427170327816849923546607148, −8.537849275213898478453834971931, −7.67689649089576688819023482713, −7.14411144116843299626082474338, −5.82744100512379228988524632303, −5.00050297673717237223578253800, −4.32890091003928348964844249072, −2.53886247059676637117516056877, 1.16204388624829845425974617776, 2.72943550601590308285914291137, 3.88995989927421758563381447032, 4.41067373592980853521322004545, 5.73437443108565207775445613130, 7.04232020523293742466767881004, 8.425931929941404703092531735781, 9.674109897354171755939637250427, 10.43717942926244986062782223754, 10.84771453041370445166790420203

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