Properties

Label 2-1155-33.32-c1-0-60
Degree $2$
Conductor $1155$
Sign $-0.947 + 0.318i$
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  − 0.805·2-s + (−0.972 − 1.43i)3-s − 1.35·4-s i·5-s + (0.783 + 1.15i)6-s + i·7-s + 2.69·8-s + (−1.10 + 2.78i)9-s + 0.805i·10-s + (−0.889 + 3.19i)11-s + (1.31 + 1.93i)12-s − 0.669i·13-s − 0.805i·14-s + (−1.43 + 0.972i)15-s + 0.530·16-s + 5.06·17-s + ⋯
L(s)  = 1  − 0.569·2-s + (−0.561 − 0.827i)3-s − 0.675·4-s − 0.447i·5-s + (0.319 + 0.471i)6-s + 0.377i·7-s + 0.954·8-s + (−0.369 + 0.929i)9-s + 0.254i·10-s + (−0.268 + 0.963i)11-s + (0.379 + 0.559i)12-s − 0.185i·13-s − 0.215i·14-s + (−0.370 + 0.251i)15-s + 0.132·16-s + 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.947 + 0.318i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4075958934\)
\(L(\frac12)\) \(\approx\) \(0.4075958934\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.972 + 1.43i)T \)
5 \( 1 + iT \)
7 \( 1 - iT \)
11 \( 1 + (0.889 - 3.19i)T \)
good2 \( 1 + 0.805T + 2T^{2} \)
13 \( 1 + 0.669iT - 13T^{2} \)
17 \( 1 - 5.06T + 17T^{2} \)
19 \( 1 + 8.50iT - 19T^{2} \)
23 \( 1 + 6.77iT - 23T^{2} \)
29 \( 1 + 2.44T + 29T^{2} \)
31 \( 1 - 5.73T + 31T^{2} \)
37 \( 1 + 4.26T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 1.79iT - 43T^{2} \)
47 \( 1 - 7.04iT - 47T^{2} \)
53 \( 1 - 9.79iT - 53T^{2} \)
59 \( 1 - 11.6iT - 59T^{2} \)
61 \( 1 + 7.58iT - 61T^{2} \)
67 \( 1 + 9.11T + 67T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 + 11.9iT - 79T^{2} \)
83 \( 1 - 6.36T + 83T^{2} \)
89 \( 1 + 8.93iT - 89T^{2} \)
97 \( 1 + 9.76T + 97T^{2} \)
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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

−9.299213142743503294664616351884, −8.605306121830590267663171883156, −7.81024383110695936301154781233, −7.14899471304668853907813523935, −6.10594962012545619938429962529, −4.96194509542408601933004676457, −4.66437685515926232459881154824, −2.80406332596196995609730338792, −1.49161490713109562953911516734, −0.27674357672433780550087008789, 1.28135740929285687083581250344, 3.50211328286538189232374906249, 3.79355492791907739285606913104, 5.25509022302838914407681286523, 5.67101748566717955667979770111, 6.88316122996760743986291982713, 8.005200791500199439258652197306, 8.484141070122585012282735097091, 9.730131728119974430620000471285, 9.992311718567017316336759542779

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