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Properties

Label 2-1155-33.32-c1-0-28
Degree $2$
Conductor $1155$
Sign $0.993 + 0.113i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  − 0.892·2-s + (0.497 − 1.65i)3-s − 1.20·4-s − i·5-s + (−0.444 + 1.48i)6-s − i·7-s + 2.85·8-s + (−2.50 − 1.65i)9-s + 0.892i·10-s + (0.586 + 3.26i)11-s + (−0.598 + 1.99i)12-s + 5.84i·13-s + 0.892i·14-s + (−1.65 − 0.497i)15-s − 0.147·16-s + 4.41·17-s + ⋯
L(s)  = 1  − 0.631·2-s + (0.287 − 0.957i)3-s − 0.601·4-s − 0.447i·5-s + (−0.181 + 0.604i)6-s − 0.377i·7-s + 1.01·8-s + (−0.834 − 0.550i)9-s + 0.282i·10-s + (0.176 + 0.984i)11-s + (−0.172 + 0.576i)12-s + 1.62i·13-s + 0.238i·14-s + (−0.428 − 0.128i)15-s − 0.0367·16-s + 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.993 + 0.113i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.993 + 0.113i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.004242323\)
\(L(\frac12)\) \(\approx\) \(1.004242323\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.497 + 1.65i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
11 \( 1 + (-0.586 - 3.26i)T \)
good2 \( 1 + 0.892T + 2T^{2} \)
13 \( 1 - 5.84iT - 13T^{2} \)
17 \( 1 - 4.41T + 17T^{2} \)
19 \( 1 - 1.11iT - 19T^{2} \)
23 \( 1 - 7.41iT - 23T^{2} \)
29 \( 1 - 9.75T + 29T^{2} \)
31 \( 1 + 0.0544T + 31T^{2} \)
37 \( 1 + 4.09T + 37T^{2} \)
41 \( 1 - 0.969T + 41T^{2} \)
43 \( 1 + 0.978iT - 43T^{2} \)
47 \( 1 + 11.0iT - 47T^{2} \)
53 \( 1 - 0.255iT - 53T^{2} \)
59 \( 1 - 1.86iT - 59T^{2} \)
61 \( 1 - 5.07iT - 61T^{2} \)
67 \( 1 - 5.79T + 67T^{2} \)
71 \( 1 - 14.9iT - 71T^{2} \)
73 \( 1 - 7.44iT - 73T^{2} \)
79 \( 1 + 7.49iT - 79T^{2} \)
83 \( 1 - 5.81T + 83T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 - 1.50T + 97T^{2} \)
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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

−9.647656867936686276796083972937, −8.888250814904363622120805611395, −8.233748220925614632886739351596, −7.34349659606425268745750199302, −6.87123084675015615493615517377, −5.56426638436293548693213885829, −4.54628591455784033401802254250, −3.61768579905475293447787797644, −1.90685925535813535021747849488, −1.11293706903558387545790610000, 0.66915995403825363012877031328, 2.78115886274599132537485060672, 3.45622549861992128131046766929, 4.66763370627989862125802868459, 5.42677029557273123140184354157, 6.35810259225953061625124450450, 7.966713035816800541067896388382, 8.173697364662440460134892556457, 9.022229641643991998794101893035, 9.800713330179497217589705656275

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