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Properties

Label 2-2366-13.12-c1-0-47
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 − 2.55·3-s − 4-s − 3.48i·5-s − 2.55i·6-s − i·7-s − i·8-s + 3.53·9-s + 3.48·10-s − 2.68i·11-s + 2.55·12-s + 14-s + 8.91i·15-s + 16-s + 5.91·17-s + 3.53i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.47·3-s − 0.5·4-s − 1.55i·5-s − 1.04i·6-s − 0.377i·7-s − 0.353i·8-s + 1.17·9-s + 1.10·10-s − 0.808i·11-s + 0.737·12-s + 0.267·14-s + 2.30i·15-s + 0.250·16-s + 1.43·17-s + 0.833i·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

\(L(1)\) \(\approx\) \(1.034456753\)
\(L(\frac12)\) \(\approx\) \(1.034456753\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.55T + 3T^{2} \)
5 \( 1 + 3.48iT - 5T^{2} \)
11 \( 1 + 2.68iT - 11T^{2} \)
17 \( 1 - 5.91T + 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 - 7.05T + 23T^{2} \)
29 \( 1 - 7.13T + 29T^{2} \)
31 \( 1 - 0.782iT - 31T^{2} \)
37 \( 1 - 7.81iT - 37T^{2} \)
41 \( 1 - 0.157iT - 41T^{2} \)
43 \( 1 - 0.330T + 43T^{2} \)
47 \( 1 + 1.60iT - 47T^{2} \)
53 \( 1 - 3.92T + 53T^{2} \)
59 \( 1 + 9.54iT - 59T^{2} \)
61 \( 1 - 15.4T + 61T^{2} \)
67 \( 1 - 0.966iT - 67T^{2} \)
71 \( 1 + 4.18iT - 71T^{2} \)
73 \( 1 + 15.0iT - 73T^{2} \)
79 \( 1 + 0.293T + 79T^{2} \)
83 \( 1 + 2.87iT - 83T^{2} \)
89 \( 1 - 5.21iT - 89T^{2} \)
97 \( 1 - 3.15iT - 97T^{2} \)
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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

−8.672792542006495525596496774874, −8.117281302597077394824727783096, −7.06886262260841845637240613583, −6.39614575235369331984711227946, −5.52792307136816280769012941882, −5.01205274151735439913073361340, −4.59368670352802763608517406624, −3.28987009150520250553867571933, −1.03896114144331397049827523426, −0.66964626989908081395653762466, 1.09206430787592549107725920774, 2.42172116358956002190916830576, 3.30342163076679949548310171808, 4.29484417030043568054034472054, 5.39761232684063413292218273367, 5.82878575586913893712865735448, 6.82504658899319828816585740751, 7.28907612031769885258441340431, 8.346122535854326888354124469945, 9.620793755519226866000772645307

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