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Properties

Label 2-1155-1155.629-c0-0-0
Degree $2$
Conductor $1155$
Sign $-0.288 - 0.957i$
Analytic cond. $0.576420$
Root an. cond. $0.759223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 + 0.309i)3-s + (−0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (−0.587 + 0.809i)7-s + (0.809 + 0.587i)9-s + (−0.309 + 0.951i)11-s + (−0.587 − 0.809i)12-s + (−1.53 − 0.5i)13-s − 0.999·15-s + (0.309 + 0.951i)16-s + (0.587 + 1.80i)17-s + (0.951 + 0.309i)20-s + (−0.809 + 0.587i)21-s + (0.809 − 0.587i)25-s + (0.587 + 0.809i)27-s + (0.951 − 0.309i)28-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s + (−0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (−0.587 + 0.809i)7-s + (0.809 + 0.587i)9-s + (−0.309 + 0.951i)11-s + (−0.587 − 0.809i)12-s + (−1.53 − 0.5i)13-s − 0.999·15-s + (0.309 + 0.951i)16-s + (0.587 + 1.80i)17-s + (0.951 + 0.309i)20-s + (−0.809 + 0.587i)21-s + (0.809 − 0.587i)25-s + (0.587 + 0.809i)27-s + (0.951 − 0.309i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.288 - 0.957i$
Analytic conductor: \(0.576420\)
Root analytic conductor: \(0.759223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :0),\ -0.288 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7165035985\)
\(L(\frac12)\) \(\approx\) \(0.7165035985\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.09178767770217860924425753598, −9.491004135543668157704877676498, −8.611271105233934055508476428770, −7.923924977809416372635523082545, −7.19862623123327320208854836635, −5.91487183755321860675598658199, −4.89457872428838829179692311409, −4.13227608881315739835096999887, −3.17722489514966219404981179723, −2.06505044501560109794587399546, 0.57527468706362142150992287407, 2.82288620142241574497513372982, 3.44668285116196004010246749892, 4.35277781949870182902974091406, 5.14699040788791945752203538846, 6.93707125048561576255786736418, 7.48032219025919987207476068707, 7.961312573725298218392398720040, 9.022292023574637785149485656836, 9.413103239993682751239385920465

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