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Properties

Label 2-1155-385.384-c1-0-12
Degree $2$
Conductor $1155$
Sign $-0.294 - 0.955i$
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  − 2.52·2-s − 3-s + 4.39·4-s + (1.63 + 1.52i)5-s + 2.52·6-s + (−2.48 + 0.905i)7-s − 6.06·8-s + 9-s + (−4.13 − 3.86i)10-s + (−2.13 − 2.53i)11-s − 4.39·12-s − 5.27i·13-s + (6.28 − 2.29i)14-s + (−1.63 − 1.52i)15-s + 6.54·16-s + 5.11i·17-s + ⋯
L(s)  = 1  − 1.78·2-s − 0.577·3-s + 2.19·4-s + (0.730 + 0.682i)5-s + 1.03·6-s + (−0.939 + 0.342i)7-s − 2.14·8-s + 0.333·9-s + (−1.30 − 1.22i)10-s + (−0.645 − 0.764i)11-s − 1.26·12-s − 1.46i·13-s + (1.68 − 0.612i)14-s + (−0.421 − 0.394i)15-s + 1.63·16-s + 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.294 - 0.955i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.294 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3606437653\)
\(L(\frac12)\) \(\approx\) \(0.3606437653\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + (-1.63 - 1.52i)T \)
7 \( 1 + (2.48 - 0.905i)T \)
11 \( 1 + (2.13 + 2.53i)T \)
good2 \( 1 + 2.52T + 2T^{2} \)
13 \( 1 + 5.27iT - 13T^{2} \)
17 \( 1 - 5.11iT - 17T^{2} \)
19 \( 1 - 4.28T + 19T^{2} \)
23 \( 1 + 1.22iT - 23T^{2} \)
29 \( 1 - 8.66iT - 29T^{2} \)
31 \( 1 + 3.16iT - 31T^{2} \)
37 \( 1 + 3.79iT - 37T^{2} \)
41 \( 1 + 5.95T + 41T^{2} \)
43 \( 1 - 7.87T + 43T^{2} \)
47 \( 1 - 6.85T + 47T^{2} \)
53 \( 1 - 12.4iT - 53T^{2} \)
59 \( 1 - 1.69iT - 59T^{2} \)
61 \( 1 + 7.32T + 61T^{2} \)
67 \( 1 + 1.64iT - 67T^{2} \)
71 \( 1 + 4.95T + 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 - 2.01iT - 79T^{2} \)
83 \( 1 + 3.14iT - 83T^{2} \)
89 \( 1 + 4.88iT - 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−10.09246803552204645080315831048, −9.329400031481268056541869766480, −8.552343219825993640138354609590, −7.61777455169807446943116620654, −6.93461504649055725470178666908, −5.84386470495552648261984040399, −5.70961404504480796689146106761, −3.30405804272423512392934170279, −2.55529423175082120095442803905, −1.07834419146518990124680202546, 0.35704016917004988240091973782, 1.60981943224583665461931538365, 2.67800439448615135525718676516, 4.46085344884622611720773234090, 5.57041044366911905794497243989, 6.61969486246688340414467700169, 7.09126339093222104724871564180, 7.968513581792160267029145908896, 9.151345694222125903751933930511, 9.565454364159774511249065114156

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