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Properties

Label 2-1155-77.76-c1-0-60
Degree $2$
Conductor $1155$
Sign $-0.727 + 0.686i$
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  + 1.29i·2-s − i·3-s + 0.335·4-s − i·5-s + 1.29·6-s + (−2.23 − 1.42i)7-s + 3.01i·8-s − 9-s + 1.29·10-s + (0.810 − 3.21i)11-s − 0.335i·12-s − 3.24·13-s + (1.83 − 2.87i)14-s − 15-s − 3.21·16-s − 7.53·17-s + ⋯
L(s)  = 1  + 0.912i·2-s − 0.577i·3-s + 0.167·4-s − 0.447i·5-s + 0.526·6-s + (−0.843 − 0.537i)7-s + 1.06i·8-s − 0.333·9-s + 0.408·10-s + (0.244 − 0.969i)11-s − 0.0967i·12-s − 0.898·13-s + (0.490 − 0.769i)14-s − 0.258·15-s − 0.804·16-s − 1.82·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.727 + 0.686i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.727 + 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3265907599\)
\(L(\frac12)\) \(\approx\) \(0.3265907599\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 1.42i)T \)
11 \( 1 + (-0.810 + 3.21i)T \)
good2 \( 1 - 1.29iT - 2T^{2} \)
13 \( 1 + 3.24T + 13T^{2} \)
17 \( 1 + 7.53T + 17T^{2} \)
19 \( 1 + 2.10T + 19T^{2} \)
23 \( 1 + 1.54T + 23T^{2} \)
29 \( 1 - 7.87iT - 29T^{2} \)
31 \( 1 - 2.42iT - 31T^{2} \)
37 \( 1 + 8.46T + 37T^{2} \)
41 \( 1 - 6.82T + 41T^{2} \)
43 \( 1 + 8.43iT - 43T^{2} \)
47 \( 1 - 2.05iT - 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 + 9.43iT - 59T^{2} \)
61 \( 1 + 8.80T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 3.80T + 73T^{2} \)
79 \( 1 + 5.78iT - 79T^{2} \)
83 \( 1 - 9.24T + 83T^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 + 2.85iT - 97T^{2} \)
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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

−8.882353043349813531960764272508, −8.807704323232801428436393035763, −7.55236499026857046501005877946, −6.93808071551708958037497384688, −6.36171974877182622577693063016, −5.50976215775096962256532179972, −4.45457626661471090119259743532, −3.12217736120499351298457404154, −1.94446488466589469286033375211, −0.12334063532624017913988116253, 2.20766575424191718536806264197, 2.61631727221126745040328340900, 3.89998468305799371095107737350, 4.56896410612117719048373628857, 6.03297739826114635755589401526, 6.69805829668709898159501406288, 7.51127332182069352729137725738, 8.941009049472157416079241295039, 9.514401964994435216824728612292, 10.18785098112319697531225649663

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