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Properties

Label 2-385-7.2-c1-0-13
Degree $2$
Conductor $385$
Sign $0.740 - 0.672i$
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  + (0.531 − 0.919i)2-s + (1.35 + 2.35i)3-s + (0.435 + 0.754i)4-s + (−0.5 + 0.866i)5-s + 2.88·6-s + (0.964 − 2.46i)7-s + 3.05·8-s + (−2.18 + 3.78i)9-s + (0.531 + 0.919i)10-s + (0.5 + 0.866i)11-s + (−1.18 + 2.04i)12-s − 2.49·13-s + (−1.75 − 2.19i)14-s − 2.71·15-s + (0.748 − 1.29i)16-s + (−2.74 − 4.75i)17-s + ⋯
L(s)  = 1  + (0.375 − 0.650i)2-s + (0.783 + 1.35i)3-s + (0.217 + 0.377i)4-s + (−0.223 + 0.387i)5-s + 1.17·6-s + (0.364 − 0.931i)7-s + 1.07·8-s + (−0.728 + 1.26i)9-s + (0.167 + 0.290i)10-s + (0.150 + 0.261i)11-s + (−0.341 + 0.591i)12-s − 0.690·13-s + (−0.468 − 0.586i)14-s − 0.700·15-s + (0.187 − 0.324i)16-s + (−0.666 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.740 - 0.672i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ 0.740 - 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08675 + 0.806491i\)
\(L(\frac12)\) \(\approx\) \(2.08675 + 0.806491i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.964 + 2.46i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.531 + 0.919i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.35 - 2.35i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 + 2.49T + 13T^{2} \)
17 \( 1 + (2.74 + 4.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.108 + 0.187i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.17 - 3.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.41T + 29T^{2} \)
31 \( 1 + (2.01 + 3.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.495 - 0.857i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 9.68T + 43T^{2} \)
47 \( 1 + (-6.02 + 10.4i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.217 + 0.376i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.02 - 5.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.54 + 4.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.192 + 0.334i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (5.72 + 9.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.67 + 8.09i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.72T + 83T^{2} \)
89 \( 1 + (7.97 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.56T + 97T^{2} \)
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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

−11.35927343599714372805751311995, −10.45977960804369728925979929709, −9.946692083313381206414266126272, −8.864464096810352650476145021145, −7.71513401682286095069666311966, −7.04108493599430770240848004765, −4.93494957297304082848990954062, −4.21570704575483437936498696585, −3.41241766818228772336186393546, −2.34814984300970183947112408301, 1.53238486595744688945418933195, 2.56352324491908032978328342710, 4.45744521363452346027745163068, 5.71166496060267287840333558924, 6.52142510006929911371774804005, 7.42032708123408911432961259449, 8.310002874819223178468687306994, 8.875097910692999061655466159079, 10.32368553508135821498662043496, 11.50758176052411993444160915463

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