Properties

Label 2-1155-385.384-c1-0-13
Degree $2$
Conductor $1155$
Sign $0.342 - 0.939i$
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  + 1.12·2-s − 3-s − 0.744·4-s + (−1.27 − 1.83i)5-s − 1.12·6-s + (−2.54 − 0.709i)7-s − 3.07·8-s + 9-s + (−1.42 − 2.05i)10-s + (2.57 + 2.09i)11-s + 0.744·12-s + 0.476i·13-s + (−2.85 − 0.794i)14-s + (1.27 + 1.83i)15-s − 1.95·16-s − 4.13i·17-s + ⋯
L(s)  = 1  + 0.792·2-s − 0.577·3-s − 0.372·4-s + (−0.569 − 0.822i)5-s − 0.457·6-s + (−0.963 − 0.268i)7-s − 1.08·8-s + 0.333·9-s + (−0.451 − 0.651i)10-s + (0.775 + 0.631i)11-s + 0.214·12-s + 0.132i·13-s + (−0.763 − 0.212i)14-s + (0.328 + 0.474i)15-s − 0.489·16-s − 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8197686497\)
\(L(\frac12)\) \(\approx\) \(0.8197686497\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + (1.27 + 1.83i)T \)
7 \( 1 + (2.54 + 0.709i)T \)
11 \( 1 + (-2.57 - 2.09i)T \)
good2 \( 1 - 1.12T + 2T^{2} \)
13 \( 1 - 0.476iT - 13T^{2} \)
17 \( 1 + 4.13iT - 17T^{2} \)
19 \( 1 - 3.40T + 19T^{2} \)
23 \( 1 - 2.30iT - 23T^{2} \)
29 \( 1 - 8.78iT - 29T^{2} \)
31 \( 1 - 7.12iT - 31T^{2} \)
37 \( 1 - 4.59iT - 37T^{2} \)
41 \( 1 + 6.75T + 41T^{2} \)
43 \( 1 + 4.78T + 43T^{2} \)
47 \( 1 - 9.14T + 47T^{2} \)
53 \( 1 - 4.46iT - 53T^{2} \)
59 \( 1 + 10.6iT - 59T^{2} \)
61 \( 1 - 2.11T + 61T^{2} \)
67 \( 1 + 11.4iT - 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 9.29iT - 73T^{2} \)
79 \( 1 + 0.726iT - 79T^{2} \)
83 \( 1 - 8.78iT - 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + 4.36T + 97T^{2} \)
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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

−9.638525187825173109668141647854, −9.389938173567197164893387469242, −8.426153759238800635346462376392, −7.12659515995644642327991580113, −6.62561025523842356149810845891, −5.30590757426116676799676451393, −4.95119026250679959126435629198, −3.89956322092722956484411155986, −3.22763329443575595826139982966, −1.10643171407449987101698722570, 0.36195470367892650258551301810, 2.65794866390291389108336587698, 3.72979648875643303413004288737, 4.11334934807375768745971261598, 5.56595434754033261821315275471, 6.13177736493532756039109045603, 6.76364409872437458975850590920, 7.922424045903500256181090112017, 8.878033962398831781617938475164, 9.764137074704673587097581972412

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