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Properties

Label 2-2366-13.12-c1-0-70
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 + 2.29·3-s − 4-s − 0.901i·5-s + 2.29i·6-s − i·7-s − i·8-s + 2.24·9-s + 0.901·10-s − 4.33i·11-s − 2.29·12-s + 14-s − 2.06i·15-s + 16-s − 5.06·17-s + 2.24i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.32·3-s − 0.5·4-s − 0.403i·5-s + 0.935i·6-s − 0.377i·7-s − 0.353i·8-s + 0.749·9-s + 0.285·10-s − 1.30i·11-s − 0.661·12-s + 0.267·14-s − 0.533i·15-s + 0.250·16-s − 1.22·17-s + 0.529i·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

\(L(1)\) \(\approx\) \(1.802009454\)
\(L(\frac12)\) \(\approx\) \(1.802009454\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.29T + 3T^{2} \)
5 \( 1 + 0.901iT - 5T^{2} \)
11 \( 1 + 4.33iT - 11T^{2} \)
17 \( 1 + 5.06T + 17T^{2} \)
19 \( 1 + 6.17iT - 19T^{2} \)
23 \( 1 + 8.45T + 23T^{2} \)
29 \( 1 + 2.19T + 29T^{2} \)
31 \( 1 + 0.873iT - 31T^{2} \)
37 \( 1 - 0.144iT - 37T^{2} \)
41 \( 1 + 3.99iT - 41T^{2} \)
43 \( 1 + 7.70T + 43T^{2} \)
47 \( 1 - 2.92iT - 47T^{2} \)
53 \( 1 - 1.69T + 53T^{2} \)
59 \( 1 - 8.54iT - 59T^{2} \)
61 \( 1 - 8.33T + 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 - 3.27iT - 71T^{2} \)
73 \( 1 + 0.539iT - 73T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 - 13.2iT - 83T^{2} \)
89 \( 1 + 7.79iT - 89T^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
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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

−8.677580036279094379572152197728, −8.262506688539268757451372263619, −7.41373251781858175852442979535, −6.63728410534566326958610460167, −5.78658954526863524494073466650, −4.73842181503288225688424182255, −3.95030430144242701740695217635, −3.11309832183322939681853298628, −2.08087394599024347551972004234, −0.46719885489695889837129210044, 1.93346090683207713893598457385, 2.17252033881166163690728478797, 3.31313531443117739378816761918, 3.99528079608259682224393142995, 4.86219205563426570051327898426, 6.06397312731594066871460429688, 7.00621831222969653245523302573, 7.889528872503139906907900670267, 8.443627767749740391317340287120, 9.159275905567354539837501498114

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