Properties

Label 2-1155-385.384-c1-0-41
Degree $2$
Conductor $1155$
Sign $0.851 + 0.524i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.851 + 0.524i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.851 + 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7478881691\)
\(L(\frac12)\) \(\approx\) \(0.7478881691\)
\(L(\frac{3}{2})\) 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

−9.487745373546458836698366902848, −8.865120244475287773657673426313, −8.157812977642148708525773388414, −7.69176028343666462322710487311, −6.34294513732868805641359563205, −5.39859753107119603953123025902, −4.49988123486603795114041458148, −3.88832097760844767446594535750, −1.73200172742888386863893781319, −0.74713588724047149833131523745, 0.847453050293357844652102070728, 2.25161540789522114698179660235, 4.06171356559862606134157110929, 4.46223865427130998772719516888, 5.63209571707939435653714158813, 7.02163730396470362111365184186, 7.26839801297976134459992469580, 8.241264619390040630155626999016, 9.076460123390178802921532646948, 9.872765897142391810138238297564

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