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Properties

Label 2-1183-13.12-c1-0-60
Degree $2$
Conductor $1183$
Sign $-0.969 + 0.246i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
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  − 2.58i·2-s + 2.93·3-s − 4.68·4-s + 1.26i·5-s − 7.60i·6-s − i·7-s + 6.94i·8-s + 5.64·9-s + 3.25·10-s − 5.67i·11-s − 13.7·12-s − 2.58·14-s + 3.70i·15-s + 8.58·16-s + 1.07·17-s − 14.5i·18-s + ⋯
L(s)  = 1  − 1.82i·2-s + 1.69·3-s − 2.34·4-s + 0.563i·5-s − 3.10i·6-s − 0.377i·7-s + 2.45i·8-s + 1.88·9-s + 1.03·10-s − 1.71i·11-s − 3.97·12-s − 0.691·14-s + 0.956i·15-s + 2.14·16-s + 0.260·17-s − 3.43i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.969 + 0.246i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.969 + 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.492081568\)
\(L(\frac12)\) \(\approx\) \(2.492081568\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + iT \)
13 \( 1 \)
good2 \( 1 + 2.58iT - 2T^{2} \)
3 \( 1 - 2.93T + 3T^{2} \)
5 \( 1 - 1.26iT - 5T^{2} \)
11 \( 1 + 5.67iT - 11T^{2} \)
17 \( 1 - 1.07T + 17T^{2} \)
19 \( 1 + 0.612iT - 19T^{2} \)
23 \( 1 + 3.02T + 23T^{2} \)
29 \( 1 - 1.64T + 29T^{2} \)
31 \( 1 + 8.21iT - 31T^{2} \)
37 \( 1 + 4.81iT - 37T^{2} \)
41 \( 1 - 0.993iT - 41T^{2} \)
43 \( 1 - 4.96T + 43T^{2} \)
47 \( 1 + 3.93iT - 47T^{2} \)
53 \( 1 - 3.04T + 53T^{2} \)
59 \( 1 - 7.61iT - 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 - 0.519iT - 71T^{2} \)
73 \( 1 - 15.9iT - 73T^{2} \)
79 \( 1 + 4.51T + 79T^{2} \)
83 \( 1 + 3.75iT - 83T^{2} \)
89 \( 1 - 16.2iT - 89T^{2} \)
97 \( 1 - 7.28iT - 97T^{2} \)
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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

−9.503284853794162203683142660637, −8.762203199605328237415980949898, −8.250391105236689223213258632238, −7.31146650361594416344968017622, −5.82845263093352915184327423430, −4.29704939873177835349714706984, −3.64189448711992148329845750974, −2.95173896527023942238396327600, −2.30630564709884041536462956066, −0.950677405967244493775029599925, 1.77787258473933947394860357206, 3.24437219420234481190429545256, 4.49439073118095731612425366411, 4.86257137633652768997246375228, 6.19995398703709355370956207761, 7.16358395944682619861171979017, 7.66459469980713556233704663439, 8.407357912730335291052514099456, 8.989337619601413717667640918158, 9.551254274772863963801758528031

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