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Properties

Label 2-1190-17.16-c1-0-26
Degree $2$
Conductor $1190$
Sign $i$
Analytic cond. $9.50219$
Root an. cond. $3.08256$
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  + 2-s − 2.56i·3-s + 4-s + i·5-s − 2.56i·6-s + i·7-s + 8-s − 3.56·9-s + i·10-s − 2i·11-s − 2.56i·12-s + 6.56·13-s + i·14-s + 2.56·15-s + 16-s − 4.12i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.47i·3-s + 0.5·4-s + 0.447i·5-s − 1.04i·6-s + 0.377i·7-s + 0.353·8-s − 1.18·9-s + 0.316i·10-s − 0.603i·11-s − 0.739i·12-s + 1.81·13-s + 0.267i·14-s + 0.661·15-s + 0.250·16-s − 0.999i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1190\)    =    \(2 \cdot 5 \cdot 7 \cdot 17\)
Sign: $i$
Analytic conductor: \(9.50219\)
Root analytic conductor: \(3.08256\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1190} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1190,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.640407376\)
\(L(\frac12)\) \(\approx\) \(2.640407376\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
17 \( 1 + 4.12iT \)
good3 \( 1 + 2.56iT - 3T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 6.56T + 13T^{2} \)
19 \( 1 + 2.56T + 19T^{2} \)
23 \( 1 + 1.12iT - 23T^{2} \)
29 \( 1 + 10.5iT - 29T^{2} \)
31 \( 1 - 2.56iT - 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 4.56T + 47T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 + 6.56T + 59T^{2} \)
61 \( 1 + 5.68iT - 61T^{2} \)
67 \( 1 + 1.12T + 67T^{2} \)
71 \( 1 + 4.56iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 - 17.3iT - 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 - 9.68iT - 97T^{2} \)
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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

−9.457956190817800280937961288035, −8.348778793717170851941898531640, −7.87619781223438644671554047719, −6.80038643661279316039393411856, −6.24286556845818955563485487175, −5.71556044284375399472767245475, −4.26117357562893845102076822830, −3.10503027241112522513237359661, −2.24276695435536326466209891274, −0.989941302245934697889475420232, 1.61110676305463983165510145897, 3.34642901637349961705305251909, 3.96581623548054619202111947468, 4.59557479469362459148177043494, 5.55031059914493872132135177392, 6.28833068565171375306092832741, 7.46389850839676972802867143591, 8.665957735603968278844448845511, 9.052946556013062631327572598903, 10.26875304991496370677181172266

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