Properties

Label 2-1183-7.4-c1-0-7
Degree $2$
Conductor $1183$
Sign $-0.999 - 0.0144i$
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  + (1.06 + 1.84i)2-s + (0.0894 − 0.154i)3-s + (−1.25 + 2.18i)4-s + (−1.80 − 3.12i)5-s + 0.380·6-s + (−2.35 − 1.20i)7-s − 1.10·8-s + (1.48 + 2.57i)9-s + (3.83 − 6.63i)10-s + (−1.99 + 3.45i)11-s + (0.225 + 0.389i)12-s + (−0.274 − 5.61i)14-s − 0.644·15-s + (1.34 + 2.33i)16-s + (−2.39 + 4.14i)17-s + (−3.15 + 5.46i)18-s + ⋯
L(s)  = 1  + (0.751 + 1.30i)2-s + (0.0516 − 0.0894i)3-s + (−0.629 + 1.09i)4-s + (−0.806 − 1.39i)5-s + 0.155·6-s + (−0.889 − 0.457i)7-s − 0.389·8-s + (0.494 + 0.856i)9-s + (1.21 − 2.09i)10-s + (−0.601 + 1.04i)11-s + (0.0649 + 0.112i)12-s + (−0.0734 − 1.50i)14-s − 0.166·15-s + (0.337 + 0.583i)16-s + (−0.580 + 1.00i)17-s + (−0.743 + 1.28i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.999 - 0.0144i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.999 - 0.0144i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.162846780\)
\(L(\frac12)\) \(\approx\) \(1.162846780\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.35 + 1.20i)T \)
13 \( 1 \)
good2 \( 1 + (-1.06 - 1.84i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.0894 + 0.154i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.80 + 3.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.99 - 3.45i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.39 - 4.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.57 - 2.72i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.08 - 1.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.57T + 29T^{2} \)
31 \( 1 + (0.743 - 1.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.48 + 4.29i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.11T + 41T^{2} \)
43 \( 1 - 1.43T + 43T^{2} \)
47 \( 1 + (0.509 + 0.882i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.01 - 5.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.45 + 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.01 - 1.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.95 - 3.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.80T + 71T^{2} \)
73 \( 1 + (1.54 - 2.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.984 + 1.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.66T + 83T^{2} \)
89 \( 1 + (-6.39 - 11.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.35T + 97T^{2} \)
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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

−10.06478488911362185682917139322, −9.066531417492969486480807717085, −8.153632200534964156881505725668, −7.53936989961550413795718759458, −7.05367024233507875833075782482, −5.84326977142604914983895175501, −5.06964677532743638195113066346, −4.34818904104435161237205389621, −3.72473491213568973140140425406, −1.69210971805597891596114822090, 0.37874756908028885619114796342, 2.43388833711818713027116516825, 3.23364420672171987626223642016, 3.51590538275297422638772386904, 4.70073862015222027722295884616, 5.90250675795896147079856754075, 6.83111088221218338255790068290, 7.49308703690057916848185227006, 8.849865345185449284562921434802, 9.701282079828290754829343244251

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