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Properties

Label 2-385-11.4-c1-0-15
Degree $2$
Conductor $385$
Sign $0.642 - 0.766i$
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  + (1.80 + 1.31i)2-s + (0.190 − 0.587i)3-s + (0.927 + 2.85i)4-s + (0.809 − 0.587i)5-s + (1.11 − 0.812i)6-s + (−0.309 − 0.951i)7-s + (−0.690 + 2.12i)8-s + (2.11 + 1.53i)9-s + 2.23·10-s + (3.04 + 1.31i)11-s + 1.85·12-s + (−3.73 − 2.71i)13-s + (0.690 − 2.12i)14-s + (−0.190 − 0.587i)15-s + (0.809 − 0.587i)16-s + (−4.54 + 3.30i)17-s + ⋯
L(s)  = 1  + (1.27 + 0.929i)2-s + (0.110 − 0.339i)3-s + (0.463 + 1.42i)4-s + (0.361 − 0.262i)5-s + (0.456 − 0.331i)6-s + (−0.116 − 0.359i)7-s + (−0.244 + 0.751i)8-s + (0.706 + 0.512i)9-s + 0.707·10-s + (0.918 + 0.396i)11-s + 0.535·12-s + (−1.03 − 0.752i)13-s + (0.184 − 0.568i)14-s + (−0.0493 − 0.151i)15-s + (0.202 − 0.146i)16-s + (−1.10 + 0.800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58798 + 1.20723i\)
\(L(\frac12)\) \(\approx\) \(2.58798 + 1.20723i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-3.04 - 1.31i)T \)
good2 \( 1 + (-1.80 - 1.31i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.190 + 0.587i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (3.73 + 2.71i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.54 - 3.30i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1 - 3.07i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 + (-0.263 - 0.812i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.61 + 1.90i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.23 + 6.88i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.763 + 2.35i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.23T + 43T^{2} \)
47 \( 1 + (0.354 - 1.08i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.23 - 0.898i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.85 + 8.78i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.85 + 2.80i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 8.18T + 67T^{2} \)
71 \( 1 + (13.2 - 9.59i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.97 - 6.06i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.73 + 6.34i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.73 - 5.62i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (-12.3 - 9.00i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.98405036108247322508353396488, −10.50716029077154583426641505139, −9.687801475476947061301530091565, −8.270869802094849645931871805198, −7.36165845490921328056032928947, −6.65369342641870247604759330922, −5.68809756656147936752614489370, −4.59901683279089008127776749930, −3.83765309896036906302585467467, −2.01095597766993901090015797377, 1.89495886094393538051348249689, 3.03707440713446615480943630104, 4.20216719096914607641336754445, 4.89409302883165162305570134465, 6.24513406308269589550518211550, 6.99920081816598284097164667608, 8.853412107739314143592732622119, 9.629245494890963965884279866192, 10.46030715164201941025445986274, 11.64518147522548667856673850920

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