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Properties

Label 2-1155-385.384-c1-0-56
Degree $2$
Conductor $1155$
Sign $0.609 + 0.793i$
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  + 0.567·2-s − 3-s − 1.67·4-s + (2.10 − 0.760i)5-s − 0.567·6-s + (2.63 − 0.228i)7-s − 2.08·8-s + 9-s + (1.19 − 0.431i)10-s + (−0.727 − 3.23i)11-s + 1.67·12-s + 4.44i·13-s + (1.49 − 0.129i)14-s + (−2.10 + 0.760i)15-s + 2.17·16-s − 3.87i·17-s + ⋯
L(s)  = 1  + 0.401·2-s − 0.577·3-s − 0.839·4-s + (0.940 − 0.340i)5-s − 0.231·6-s + (0.996 − 0.0863i)7-s − 0.737·8-s + 0.333·9-s + (0.377 − 0.136i)10-s + (−0.219 − 0.975i)11-s + 0.484·12-s + 1.23i·13-s + (0.399 − 0.0346i)14-s + (−0.542 + 0.196i)15-s + 0.543·16-s − 0.939i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.609 + 0.793i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.609 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.630763124\)
\(L(\frac12)\) \(\approx\) \(1.630763124\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 + 0.760i)T \)
7 \( 1 + (-2.63 + 0.228i)T \)
11 \( 1 + (0.727 + 3.23i)T \)
good2 \( 1 - 0.567T + 2T^{2} \)
13 \( 1 - 4.44iT - 13T^{2} \)
17 \( 1 + 3.87iT - 17T^{2} \)
19 \( 1 - 0.293T + 19T^{2} \)
23 \( 1 + 1.67iT - 23T^{2} \)
29 \( 1 + 0.242iT - 29T^{2} \)
31 \( 1 - 6.28iT - 31T^{2} \)
37 \( 1 + 11.3iT - 37T^{2} \)
41 \( 1 + 5.68T + 41T^{2} \)
43 \( 1 - 7.05T + 43T^{2} \)
47 \( 1 - 2.92T + 47T^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 + 5.48iT - 59T^{2} \)
61 \( 1 - 5.57T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 4.76iT - 73T^{2} \)
79 \( 1 - 7.37iT - 79T^{2} \)
83 \( 1 - 12.9iT - 83T^{2} \)
89 \( 1 + 13.0iT - 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−9.527425585823733755194826773102, −8.966647155075330842069190465057, −8.237789030457864749084700201944, −6.99968572361128031011552673355, −6.04173720355646735689776528928, −5.20783935664158954890282014301, −4.80662140187153633687677083118, −3.71962727413254790215095857963, −2.17191656286779814138108557642, −0.791534821166768284562223056824, 1.29913661474532839339820556944, 2.62995728295527263985869998025, 4.01720782604331953334572221901, 4.94928326333863037787111281780, 5.52320468031340756354869604424, 6.23223511801950194474031805569, 7.47451819307593729570017240119, 8.243840063151397320433647653607, 9.205475143472743316506111978324, 10.15560821644489332773800903376

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