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Properties

Label 2-2366-13.12-c1-0-56
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 + 3.34·3-s − 4-s + 1.56i·5-s + 3.34i·6-s − i·7-s − i·8-s + 8.18·9-s − 1.56·10-s + 2.86i·11-s − 3.34·12-s + 14-s + 5.22i·15-s + 16-s + 2.22·17-s + 8.18i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.93·3-s − 0.5·4-s + 0.699i·5-s + 1.36i·6-s − 0.377i·7-s − 0.353i·8-s + 2.72·9-s − 0.494·10-s + 0.864i·11-s − 0.965·12-s + 0.267·14-s + 1.35i·15-s + 0.250·16-s + 0.540·17-s + 1.92i·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

\(L(1)\) \(\approx\) \(3.732739869\)
\(L(\frac12)\) \(\approx\) \(3.732739869\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.34T + 3T^{2} \)
5 \( 1 - 1.56iT - 5T^{2} \)
11 \( 1 - 2.86iT - 11T^{2} \)
17 \( 1 - 2.22T + 17T^{2} \)
19 \( 1 + 7.23iT - 19T^{2} \)
23 \( 1 - 1.66T + 23T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + 0.597iT - 31T^{2} \)
37 \( 1 + 0.0385iT - 37T^{2} \)
41 \( 1 - 7.95iT - 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 7.02iT - 47T^{2} \)
53 \( 1 + 5.98T + 53T^{2} \)
59 \( 1 - 0.896iT - 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 - 1.64iT - 67T^{2} \)
71 \( 1 + 2.29iT - 71T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 - 4.26T + 79T^{2} \)
83 \( 1 + 4.94iT - 83T^{2} \)
89 \( 1 + 2.42iT - 89T^{2} \)
97 \( 1 + 4.88iT - 97T^{2} \)
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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

−9.045173383973337820254348448691, −8.246752765615955366028492571493, −7.63613975529296236284647071677, −6.97208893212934821742637716390, −6.56575324605820833005178966439, −4.84571870813974388787381224248, −4.35968406373433591034895900829, −3.16666907709086992282922284898, −2.75767835067011089940151154748, −1.42283247614129531685830818695, 1.19613903470974563896250555108, 2.01000148615358992653645545767, 3.12922542856450717901637804147, 3.52680157627976291316565948654, 4.50587182708865069899432896339, 5.42318389656732978438018787941, 6.64460699015099268542264481461, 7.84671184623907950613199083689, 8.242482502169434364775952711522, 8.832500830060239259887084064618

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