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Properties

Label 2-2366-13.12-c1-0-30
Degree $2$
Conductor $2366$
Sign $0.277 - 0.960i$
Analytic cond. $18.8926$
Root an. cond. $4.34656$
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  + i·2-s + 0.252·3-s − 4-s + 1.14i·5-s + 0.252i·6-s − i·7-s − i·8-s − 2.93·9-s − 1.14·10-s − 4.44i·11-s − 0.252·12-s + 14-s + 0.290i·15-s + 16-s − 2.70·17-s − 2.93i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.145·3-s − 0.5·4-s + 0.513i·5-s + 0.103i·6-s − 0.377i·7-s − 0.353i·8-s − 0.978·9-s − 0.362·10-s − 1.34i·11-s − 0.0729·12-s + 0.267·14-s + 0.0749i·15-s + 0.250·16-s − 0.657·17-s − 0.692i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2366\)    =    \(2 \cdot 7 \cdot 13^{2}\)
Sign: $0.277 - 0.960i$
Analytic conductor: \(18.8926\)
Root analytic conductor: \(4.34656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2366} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2366,\ (\ :1/2),\ 0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.487364340\)
\(L(\frac12)\) \(\approx\) \(1.487364340\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 + iT \)
13 \( 1 \)
good3 \( 1 - 0.252T + 3T^{2} \)
5 \( 1 - 1.14iT - 5T^{2} \)
11 \( 1 + 4.44iT - 11T^{2} \)
17 \( 1 + 2.70T + 17T^{2} \)
19 \( 1 - 6.56iT - 19T^{2} \)
23 \( 1 - 2.07T + 23T^{2} \)
29 \( 1 - 7.19T + 29T^{2} \)
31 \( 1 + 7.90iT - 31T^{2} \)
37 \( 1 - 9.64iT - 37T^{2} \)
41 \( 1 - 9.49iT - 41T^{2} \)
43 \( 1 - 3.40T + 43T^{2} \)
47 \( 1 - 1.67iT - 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 - 0.0677iT - 59T^{2} \)
61 \( 1 - 8.10T + 61T^{2} \)
67 \( 1 - 0.513iT - 67T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 - 8.02iT - 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 + 10.8iT - 89T^{2} \)
97 \( 1 - 2.12iT - 97T^{2} \)
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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

−8.771828717724474106205928485235, −8.371395890063825378612763241356, −7.71327101718496921864712108599, −6.64097908299546008710956503536, −6.16167846785977938850205395541, −5.43051391438848818127626991087, −4.35084016915458352585334712949, −3.39370992819842944214358951443, −2.65768752489308381071397043541, −0.882344115169576746490657376685, 0.67718520280646676076604458040, 2.17305803949644031019977878960, 2.71657066374075656393025839624, 3.92183269870989059169814943254, 4.93245947579326074076993547009, 5.24849792917729782556914882601, 6.59132564536114227013983556530, 7.26651699099109485456105482764, 8.424596732711086745308792421808, 9.003113195907032129896644344284

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