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Properties

Label 2-385-7.4-c1-0-7
Degree $2$
Conductor $385$
Sign $-0.371 - 0.928i$
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  + (0.323 + 0.559i)2-s + (−1.03 + 1.79i)3-s + (0.790 − 1.37i)4-s + (0.5 + 0.866i)5-s − 1.34·6-s + (−0.211 + 2.63i)7-s + 2.31·8-s + (−0.655 − 1.13i)9-s + (−0.323 + 0.559i)10-s + (0.5 − 0.866i)11-s + (1.64 + 2.84i)12-s − 0.398·13-s + (−1.54 + 0.734i)14-s − 2.07·15-s + (−0.833 − 1.44i)16-s + (−2.20 + 3.82i)17-s + ⋯
L(s)  = 1  + (0.228 + 0.395i)2-s + (−0.599 + 1.03i)3-s + (0.395 − 0.685i)4-s + (0.223 + 0.387i)5-s − 0.548·6-s + (−0.0798 + 0.996i)7-s + 0.818·8-s + (−0.218 − 0.378i)9-s + (−0.102 + 0.177i)10-s + (0.150 − 0.261i)11-s + (0.474 + 0.821i)12-s − 0.110·13-s + (−0.412 + 0.196i)14-s − 0.536·15-s + (−0.208 − 0.360i)16-s + (−0.535 + 0.927i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.371 - 0.928i$
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: \((2,\ 385,\ (\ :1/2),\ -0.371 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.792997 + 1.17100i\)
\(L(\frac12)\) \(\approx\) \(0.792997 + 1.17100i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.211 - 2.63i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.323 - 0.559i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.03 - 1.79i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 + 0.398T + 13T^{2} \)
17 \( 1 + (2.20 - 3.82i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.42 - 2.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.24 - 3.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.69T + 29T^{2} \)
31 \( 1 + (-1.17 + 2.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.24 + 5.62i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.10T + 41T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 + (-3.78 - 6.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.46 + 5.99i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.32 + 5.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.914 - 1.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.61 + 4.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + (-8.15 + 14.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.03 - 3.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + (0.183 + 0.317i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−11.20254860459599246178402114286, −10.85871694625302366044545595971, −9.848368685262364120286633996354, −9.205995203626760925644585622818, −7.78023200796668739712316715567, −6.48667738603542736134461127000, −5.72401907687807771099289473875, −5.11036249455886125535252083138, −3.78758361257509311763870615329, −2.07295160807309207473014581961, 0.995647014130005702310698930167, 2.45555861638223335311780875619, 3.96831079163402529571304988196, 5.09043586203941912488096918948, 6.70535260805226716222884530107, 7.02585339065082168661896277905, 7.978417996203134659499762581590, 9.234073793853424271492972354585, 10.46258491244018761920459346105, 11.32078712626428744179810656022

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