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Properties

Label 2-1155-385.384-c1-0-68
Degree $2$
Conductor $1155$
Sign $-0.534 + 0.844i$
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  − 1.69·2-s + 3-s + 0.874·4-s + (−2.22 − 0.217i)5-s − 1.69·6-s + (0.698 − 2.55i)7-s + 1.90·8-s + 9-s + (3.77 + 0.368i)10-s + (3.25 + 0.658i)11-s + 0.874·12-s − 5.21i·13-s + (−1.18 + 4.32i)14-s + (−2.22 − 0.217i)15-s − 4.98·16-s − 3.07i·17-s + ⋯
L(s)  = 1  − 1.19·2-s + 0.577·3-s + 0.437·4-s + (−0.995 − 0.0972i)5-s − 0.692·6-s + (0.263 − 0.964i)7-s + 0.674·8-s + 0.333·9-s + (1.19 + 0.116i)10-s + (0.980 + 0.198i)11-s + 0.252·12-s − 1.44i·13-s + (−0.316 + 1.15i)14-s + (−0.574 − 0.0561i)15-s − 1.24·16-s − 0.746i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.534 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.534 + 0.844i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.534 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6335855230\)
\(L(\frac12)\) \(\approx\) \(0.6335855230\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.217i)T \)
7 \( 1 + (-0.698 + 2.55i)T \)
11 \( 1 + (-3.25 - 0.658i)T \)
good2 \( 1 + 1.69T + 2T^{2} \)
13 \( 1 + 5.21iT - 13T^{2} \)
17 \( 1 + 3.07iT - 17T^{2} \)
19 \( 1 + 4.27T + 19T^{2} \)
23 \( 1 - 6.74iT - 23T^{2} \)
29 \( 1 + 0.101iT - 29T^{2} \)
31 \( 1 - 6.52iT - 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 + 6.80T + 43T^{2} \)
47 \( 1 + 6.68T + 47T^{2} \)
53 \( 1 + 2.30iT - 53T^{2} \)
59 \( 1 + 7.18iT - 59T^{2} \)
61 \( 1 + 4.40T + 61T^{2} \)
67 \( 1 + 4.25iT - 67T^{2} \)
71 \( 1 + 5.83T + 71T^{2} \)
73 \( 1 + 8.09iT - 73T^{2} \)
79 \( 1 + 14.2iT - 79T^{2} \)
83 \( 1 - 7.09iT - 83T^{2} \)
89 \( 1 + 16.3iT - 89T^{2} \)
97 \( 1 - 2.98T + 97T^{2} \)
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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

−9.329488399436684988784768495833, −8.735568896932181728538321791467, −7.74859895550461023759657385820, −7.61160600374983941225013870034, −6.68483808581902756614085586740, −5.00722331470287192414349535978, −4.10306515531367673746083649899, −3.28394926557310788400960036939, −1.56119358150963487914156805752, −0.42746661013302209921123472244, 1.44526091586099585702888888847, 2.54067799115209074484522642767, 4.09674582992804128321912010051, 4.51067234745244512179037946694, 6.33242539501353650604198972556, 6.92379079222806805443465707171, 8.138296589830743926018397315716, 8.427071868328207860739655218651, 9.055985558128490510756642732021, 9.745876680904217131370708174424

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