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Properties

Label 2-1155-5.4-c1-0-43
Degree $2$
Conductor $1155$
Sign $0.489 + 0.871i$
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  + 1.76i·2-s + i·3-s − 1.13·4-s + (−1.09 − 1.94i)5-s − 1.76·6-s − i·7-s + 1.53i·8-s − 9-s + (3.44 − 1.93i)10-s − 11-s − 1.13i·12-s + 2.21i·13-s + 1.76·14-s + (1.94 − 1.09i)15-s − 4.98·16-s − 7.13i·17-s + ⋯
L(s)  = 1  + 1.25i·2-s + 0.577i·3-s − 0.565·4-s + (−0.489 − 0.871i)5-s − 0.722·6-s − 0.377i·7-s + 0.543i·8-s − 0.333·9-s + (1.09 − 0.612i)10-s − 0.301·11-s − 0.326i·12-s + 0.615i·13-s + 0.472·14-s + (0.503 − 0.282i)15-s − 1.24·16-s − 1.73i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.489 + 0.871i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.489 + 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2921856782\)
\(L(\frac12)\) \(\approx\) \(0.2921856782\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 + 1.94i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
good2 \( 1 - 1.76iT - 2T^{2} \)
13 \( 1 - 2.21iT - 13T^{2} \)
17 \( 1 + 7.13iT - 17T^{2} \)
19 \( 1 + 7.91T + 19T^{2} \)
23 \( 1 + 1.96iT - 23T^{2} \)
29 \( 1 + 5.96T + 29T^{2} \)
31 \( 1 + 1.40T + 31T^{2} \)
37 \( 1 + 1.37iT - 37T^{2} \)
41 \( 1 - 4.80T + 41T^{2} \)
43 \( 1 + 2.34iT - 43T^{2} \)
47 \( 1 + 9.10iT - 47T^{2} \)
53 \( 1 + 2.65iT - 53T^{2} \)
59 \( 1 + 6.97T + 59T^{2} \)
61 \( 1 + 1.37T + 61T^{2} \)
67 \( 1 - 12.8iT - 67T^{2} \)
71 \( 1 - 1.16T + 71T^{2} \)
73 \( 1 + 4.94iT - 73T^{2} \)
79 \( 1 + 0.590T + 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 - 9.79T + 89T^{2} \)
97 \( 1 - 12.8iT - 97T^{2} \)
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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

−9.162209638177711889175157239247, −8.870967876182031712883047848607, −7.889701013977492531086240558347, −7.24505060163874364280417142215, −6.35962939009201268633174163792, −5.31932404359394357155734511377, −4.69897177864049910902478570547, −3.90735104178632967186605756484, −2.27730501150165928181133481970, −0.11722955269648599342360078435, 1.69986064082182004813237235273, 2.54265037146672774663978073449, 3.47351140620896419229291943484, 4.30196524196503862205657833863, 5.93262773018149969549633357568, 6.50257661477750017884464677997, 7.60457327709534530771566271713, 8.266733303104854470277017846627, 9.285918022833138655479057856517, 10.34093022251524331723386338666

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