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Properties

Label 2-1155-33.32-c1-0-79
Degree $2$
Conductor $1155$
Sign $0.999 - 0.0100i$
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  + 2.21·2-s + (1.64 + 0.539i)3-s + 2.90·4-s − i·5-s + (3.64 + 1.19i)6-s − i·7-s + 2.00·8-s + (2.41 + 1.77i)9-s − 2.21i·10-s + (3.14 − 1.06i)11-s + (4.77 + 1.56i)12-s + 2.21i·13-s − 2.21i·14-s + (0.539 − 1.64i)15-s − 1.37·16-s − 2.72·17-s + ⋯
L(s)  = 1  + 1.56·2-s + (0.950 + 0.311i)3-s + 1.45·4-s − 0.447i·5-s + (1.48 + 0.487i)6-s − 0.377i·7-s + 0.707·8-s + (0.806 + 0.591i)9-s − 0.700i·10-s + (0.947 − 0.320i)11-s + (1.37 + 0.452i)12-s + 0.613i·13-s − 0.591i·14-s + (0.139 − 0.424i)15-s − 0.343·16-s − 0.660·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.999 - 0.0100i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.999 - 0.0100i)\)

Particular Values

\(L(1)\) \(\approx\) \(5.313325318\)
\(L(\frac12)\) \(\approx\) \(5.313325318\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 - 0.539i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
11 \( 1 + (-3.14 + 1.06i)T \)
good2 \( 1 - 2.21T + 2T^{2} \)
13 \( 1 - 2.21iT - 13T^{2} \)
17 \( 1 + 2.72T + 17T^{2} \)
19 \( 1 - 2.83iT - 19T^{2} \)
23 \( 1 + 5.72iT - 23T^{2} \)
29 \( 1 + 1.05T + 29T^{2} \)
31 \( 1 - 7.54T + 31T^{2} \)
37 \( 1 + 5.29T + 37T^{2} \)
41 \( 1 + 9.83T + 41T^{2} \)
43 \( 1 - 5.48iT - 43T^{2} \)
47 \( 1 + 4.94iT - 47T^{2} \)
53 \( 1 + 10.4iT - 53T^{2} \)
59 \( 1 - 14.4iT - 59T^{2} \)
61 \( 1 - 4.24iT - 61T^{2} \)
67 \( 1 - 2.41T + 67T^{2} \)
71 \( 1 - 2.12iT - 71T^{2} \)
73 \( 1 + 3.61iT - 73T^{2} \)
79 \( 1 - 1.37iT - 79T^{2} \)
83 \( 1 - 1.18T + 83T^{2} \)
89 \( 1 - 0.596iT - 89T^{2} \)
97 \( 1 + 3.76T + 97T^{2} \)
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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

−9.803197835099744924292637119224, −8.842058419429138965697187182111, −8.289104250331785815733722457007, −6.94027185539182003546189165562, −6.45213241357744596566352926365, −5.19863942972273043065309908907, −4.32311274898503131910982511999, −3.90792771562804009485366647609, −2.86785201183379444106602439697, −1.70727254480124329678249432365, 1.81235659760341257242976812217, 2.84572033499919220801617810394, 3.54569796722946109541389886182, 4.42737061250566375471904016204, 5.41863424027127385107412518818, 6.54492676632091458248259467417, 6.90931940952125755668665882553, 8.001588418427887550627176060023, 8.986635705848445908492617503145, 9.688209941361984660923886795331

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