Properties

Label 2-1155-77.76-c1-0-24
Degree $2$
Conductor $1155$
Sign $0.954 + 0.299i$
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  − 0.792i·2-s i·3-s + 1.37·4-s + i·5-s − 0.792·6-s + (−0.792 + 2.52i)7-s − 2.67i·8-s − 9-s + 0.792·10-s + 3.31i·11-s − 1.37i·12-s + (2 + 0.627i)14-s + 15-s + 0.627·16-s + 4.10·17-s + 0.792i·18-s + ⋯
L(s)  = 1  − 0.560i·2-s − 0.577i·3-s + 0.686·4-s + 0.447i·5-s − 0.323·6-s + (−0.299 + 0.954i)7-s − 0.944i·8-s − 0.333·9-s + 0.250·10-s + 1.00i·11-s − 0.396i·12-s + (0.534 + 0.167i)14-s + 0.258·15-s + 0.156·16-s + 0.996·17-s + 0.186i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.954 + 0.299i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.954 + 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.959788746\)
\(L(\frac12)\) \(\approx\) \(1.959788746\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 - iT \)
7 \( 1 + (0.792 - 2.52i)T \)
11 \( 1 - 3.31iT \)
good2 \( 1 + 0.792iT - 2T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 4.10T + 17T^{2} \)
19 \( 1 + 4.40T + 19T^{2} \)
23 \( 1 - 9.11T + 23T^{2} \)
29 \( 1 - 5.98iT - 29T^{2} \)
31 \( 1 + 6.74iT - 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 - 4.10iT - 43T^{2} \)
47 \( 1 - 12.7iT - 47T^{2} \)
53 \( 1 + 1.62T + 53T^{2} \)
59 \( 1 - 1.62iT - 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 4.74T + 67T^{2} \)
71 \( 1 - 3.25T + 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + 0.294iT - 79T^{2} \)
83 \( 1 + 9.45T + 83T^{2} \)
89 \( 1 + 8.37iT - 89T^{2} \)
97 \( 1 - 8.37iT - 97T^{2} \)
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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

−9.689353894208461392183008383826, −9.225949501494232240575865377819, −7.894374904957657000767079683714, −7.27963153165073579265763950493, −6.41063525545116294148447969656, −5.79383626589021402094273522979, −4.44120281988975934742965116865, −2.99806500440245206801233574842, −2.52786415122708487258114392561, −1.33851787351539540582707758707, 0.957187527536910719350204287617, 2.72623034965580240768414812452, 3.68933993402173404153773691524, 4.76872191063629861759604731761, 5.70975649348881083359645252525, 6.44149241458695279424714464050, 7.36279788456547766003765403082, 8.128099424767217500183714952439, 8.906422994103390523326914921816, 9.846573118627587886167112385170

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