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Properties

Label 2-1155-33.32-c1-0-69
Degree $2$
Conductor $1155$
Sign $0.708 - 0.705i$
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  + 2.45·2-s + (0.877 + 1.49i)3-s + 4.00·4-s − i·5-s + (2.15 + 3.66i)6-s + i·7-s + 4.91·8-s + (−1.46 + 2.62i)9-s − 2.45i·10-s + (0.828 + 3.21i)11-s + (3.51 + 5.98i)12-s − 6.76i·13-s + 2.45i·14-s + (1.49 − 0.877i)15-s + 4.03·16-s + 5.22·17-s + ⋯
L(s)  = 1  + 1.73·2-s + (0.506 + 0.862i)3-s + 2.00·4-s − 0.447i·5-s + (0.877 + 1.49i)6-s + 0.377i·7-s + 1.73·8-s + (−0.486 + 0.873i)9-s − 0.775i·10-s + (0.249 + 0.968i)11-s + (1.01 + 1.72i)12-s − 1.87i·13-s + 0.655i·14-s + (0.385 − 0.226i)15-s + 1.00·16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.708 - 0.705i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.708 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(5.254823827\)
\(L(\frac12)\) \(\approx\) \(5.254823827\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.877 - 1.49i)T \)
5 \( 1 + iT \)
7 \( 1 - iT \)
11 \( 1 + (-0.828 - 3.21i)T \)
good2 \( 1 - 2.45T + 2T^{2} \)
13 \( 1 + 6.76iT - 13T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 - 6.21iT - 19T^{2} \)
23 \( 1 + 3.22iT - 23T^{2} \)
29 \( 1 + 4.08T + 29T^{2} \)
31 \( 1 - 2.16T + 31T^{2} \)
37 \( 1 + 8.03T + 37T^{2} \)
41 \( 1 - 5.40T + 41T^{2} \)
43 \( 1 + 6.96iT - 43T^{2} \)
47 \( 1 + 7.86iT - 47T^{2} \)
53 \( 1 - 7.52iT - 53T^{2} \)
59 \( 1 + 4.90iT - 59T^{2} \)
61 \( 1 - 5.13iT - 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 8.09iT - 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 4.27T + 83T^{2} \)
89 \( 1 - 7.30iT - 89T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−10.30582156775803800066486759371, −9.141551552916563276840714615820, −8.066530076030765858283505336584, −7.43095714651153483016718144283, −5.92382821351822120763145057347, −5.45826384024710174172200030824, −4.73609172863667625567516458250, −3.73947936371533714243349751931, −3.14860028293820763737741403266, −1.98541415564665498392917927595, 1.51151571605611967989081130564, 2.71381086645343625743307496636, 3.47384307484404685069227750351, 4.29017626189880998137249313021, 5.50203042932498233345262264103, 6.34625841645727827739050546965, 6.93479570993678415635383372352, 7.57239500859970918354929678253, 8.797443474928160895213701076001, 9.642478882957613665201850400040

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