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Properties

Label 2-1155-5.4-c1-0-31
Degree $2$
Conductor $1155$
Sign $0.807 - 0.589i$
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  + 2.51i·2-s + i·3-s − 4.30·4-s + (−1.80 + 1.31i)5-s − 2.51·6-s − i·7-s − 5.77i·8-s − 9-s + (−3.30 − 4.53i)10-s − 11-s − 4.30i·12-s − 7.07i·13-s + 2.51·14-s + (−1.31 − 1.80i)15-s + 5.89·16-s + 0.968i·17-s + ⋯
L(s)  = 1  + 1.77i·2-s + 0.577i·3-s − 2.15·4-s + (−0.807 + 0.589i)5-s − 1.02·6-s − 0.377i·7-s − 2.04i·8-s − 0.333·9-s + (−1.04 − 1.43i)10-s − 0.301·11-s − 1.24i·12-s − 1.96i·13-s + 0.670·14-s + (−0.340 − 0.466i)15-s + 1.47·16-s + 0.234i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.807 - 0.589i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.807 - 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4901691836\)
\(L(\frac12)\) \(\approx\) \(0.4901691836\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 - 1.31i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
good2 \( 1 - 2.51iT - 2T^{2} \)
13 \( 1 + 7.07iT - 13T^{2} \)
17 \( 1 - 0.968iT - 17T^{2} \)
19 \( 1 - 2.31T + 19T^{2} \)
23 \( 1 - 5.12iT - 23T^{2} \)
29 \( 1 + 6.98T + 29T^{2} \)
31 \( 1 - 6.96T + 31T^{2} \)
37 \( 1 - 4.39iT - 37T^{2} \)
41 \( 1 + 8.89T + 41T^{2} \)
43 \( 1 + 12.7iT - 43T^{2} \)
47 \( 1 + 7.09iT - 47T^{2} \)
53 \( 1 + 5.21iT - 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 + 1.53T + 61T^{2} \)
67 \( 1 - 3.70iT - 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 1.94iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 8.06iT - 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 - 4.07iT - 97T^{2} \)
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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

−9.840774069974778193400827536046, −8.622784666708842267822075784494, −8.007157431917367863213581000995, −7.47861515817667052244449074933, −6.67253933872921234640520718239, −5.57657104196766965844901045719, −5.11069158825390534784820277298, −3.90208316661413577899396465157, −3.19517321745655297601057489676, −0.24118196483591737243992979579, 1.22348841234654297872399780261, 2.21480344832851838642909898581, 3.26576980625663585557170726308, 4.34693858247374175397089504420, 4.88595998452500852120482766587, 6.32285946966414950574253869556, 7.47805082499123133592642690672, 8.418802915485619064075448725775, 9.122638611997092462935237017228, 9.639653030088523072812047718610

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