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Properties

Label 2-1155-77.76-c1-0-23
Degree $2$
Conductor $1155$
Sign $0.801 - 0.598i$
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  + 0.859i·2-s − i·3-s + 1.26·4-s − i·5-s + 0.859·6-s + (−2.42 + 1.05i)7-s + 2.80i·8-s − 9-s + 0.859·10-s + (−3.22 + 0.765i)11-s − 1.26i·12-s + 3.49·13-s + (−0.904 − 2.08i)14-s − 15-s + 0.110·16-s + 5.22·17-s + ⋯
L(s)  = 1  + 0.608i·2-s − 0.577i·3-s + 0.630·4-s − 0.447i·5-s + 0.351·6-s + (−0.917 + 0.397i)7-s + 0.991i·8-s − 0.333·9-s + 0.271·10-s + (−0.972 + 0.230i)11-s − 0.363i·12-s + 0.968·13-s + (−0.241 − 0.557i)14-s − 0.258·15-s + 0.0275·16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.801 - 0.598i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.801 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.845781664\)
\(L(\frac12)\) \(\approx\) \(1.845781664\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.42 - 1.05i)T \)
11 \( 1 + (3.22 - 0.765i)T \)
good2 \( 1 - 0.859iT - 2T^{2} \)
13 \( 1 - 3.49T + 13T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 - 4.83T + 19T^{2} \)
23 \( 1 - 3.63T + 23T^{2} \)
29 \( 1 - 6.88iT - 29T^{2} \)
31 \( 1 - 1.77iT - 31T^{2} \)
37 \( 1 - 9.57T + 37T^{2} \)
41 \( 1 + 2.91T + 41T^{2} \)
43 \( 1 + 4.10iT - 43T^{2} \)
47 \( 1 + 6.68iT - 47T^{2} \)
53 \( 1 - 7.05T + 53T^{2} \)
59 \( 1 + 4.22iT - 59T^{2} \)
61 \( 1 - 3.39T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + 7.34T + 73T^{2} \)
79 \( 1 - 6.63iT - 79T^{2} \)
83 \( 1 + 7.78T + 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
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

−9.824520214692872857624505878851, −8.812505478441427467440405776834, −8.097530105682814879153490632048, −7.30462082783669894533847221016, −6.65867666733679388382637708567, −5.58241060661327628494125126412, −5.33995925160888369124873192187, −3.43999592477476780226318961436, −2.63379320405771352961265051576, −1.20523750108851127264743960995, 0.956782024917900070070633092183, 2.76679800132636399071215434700, 3.20696400727504540502539602144, 4.13854400890804090542213398675, 5.64512987116812798200404150625, 6.20401161731599141750069250901, 7.31466832663438611513397478250, 7.930160477653063043357889677233, 9.270269574255503907179629189649, 10.05953999266562718043550711628

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