Properties

Label 2-1155-77.76-c1-0-28
Degree $2$
Conductor $1155$
Sign $0.0403 - 0.999i$
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  + 0.474i·2-s + i·3-s + 1.77·4-s i·5-s − 0.474·6-s + (0.474 + 2.60i)7-s + 1.79i·8-s − 9-s + 0.474·10-s + (3.23 + 0.726i)11-s + 1.77i·12-s − 0.814·13-s + (−1.23 + 0.225i)14-s + 15-s + 2.69·16-s + 6.40·17-s + ⋯
L(s)  = 1  + 0.335i·2-s + 0.577i·3-s + 0.887·4-s − 0.447i·5-s − 0.193·6-s + (0.179 + 0.983i)7-s + 0.633i·8-s − 0.333·9-s + 0.150·10-s + (0.975 + 0.219i)11-s + 0.512i·12-s − 0.225·13-s + (−0.330 + 0.0602i)14-s + 0.258·15-s + 0.674·16-s + 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0403 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0403 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0403 - 0.999i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.0403 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.166357519\)
\(L(\frac12)\) \(\approx\) \(2.166357519\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.474 - 2.60i)T \)
11 \( 1 + (-3.23 - 0.726i)T \)
good2 \( 1 - 0.474iT - 2T^{2} \)
13 \( 1 + 0.814T + 13T^{2} \)
17 \( 1 - 6.40T + 17T^{2} \)
19 \( 1 + 4.95T + 19T^{2} \)
23 \( 1 + 0.774T + 23T^{2} \)
29 \( 1 - 2.51iT - 29T^{2} \)
31 \( 1 + 5.42iT - 31T^{2} \)
37 \( 1 + 5.42T + 37T^{2} \)
41 \( 1 - 2.77T + 41T^{2} \)
43 \( 1 - 6.55iT - 43T^{2} \)
47 \( 1 - 4.57iT - 47T^{2} \)
53 \( 1 - 9.63T + 53T^{2} \)
59 \( 1 + 3.34iT - 59T^{2} \)
61 \( 1 + 8.03T + 61T^{2} \)
67 \( 1 - 16.0T + 67T^{2} \)
71 \( 1 + 3.04T + 71T^{2} \)
73 \( 1 + 9.23T + 73T^{2} \)
79 \( 1 + 7.97iT - 79T^{2} \)
83 \( 1 - 6.58T + 83T^{2} \)
89 \( 1 - 1.12iT - 89T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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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

−9.902453412731272772805848301478, −9.153352714257387853816676378066, −8.352820276906183139173482560161, −7.61783476044990301564659382112, −6.47781655371294803007273764818, −5.81057440309018851296656617305, −5.03368860861312085141767889943, −3.88646980141058450540465285006, −2.72839039123051453017769031601, −1.60548767575954403437744871745, 1.00863072030114355832978182673, 2.04103866604268116015605706662, 3.28142841712379125061014864284, 4.01205462521742132049066568238, 5.55261540069626303493626919156, 6.51117981155083342869757422923, 7.04503022397032173151824268800, 7.72485167879666237953314448657, 8.662951478205005080140287496140, 9.945066628190083852783756706755

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