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Properties

Label 2-1183-7.4-c1-0-83
Degree $2$
Conductor $1183$
Sign $-0.902 + 0.431i$
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  + (−0.00118 − 0.00205i)2-s + (1.54 − 2.68i)3-s + (0.999 − 1.73i)4-s + (0.565 + 0.979i)5-s − 0.00733·6-s + (−2.64 − 0.0383i)7-s − 0.00946·8-s + (−3.29 − 5.71i)9-s + (0.00133 − 0.00231i)10-s + (2.23 − 3.87i)11-s + (−3.09 − 5.36i)12-s + (0.00305 + 0.00546i)14-s + 3.50·15-s + (−1.99 − 3.46i)16-s + (−1.94 + 3.36i)17-s + (−0.00781 + 0.0135i)18-s + ⋯
L(s)  = 1  + (−0.000836 − 0.00144i)2-s + (0.894 − 1.54i)3-s + (0.499 − 0.866i)4-s + (0.252 + 0.438i)5-s − 0.00299·6-s + (−0.999 − 0.0144i)7-s − 0.00334·8-s + (−1.09 − 1.90i)9-s + (0.000423 − 0.000733i)10-s + (0.675 − 1.16i)11-s + (−0.894 − 1.54i)12-s + (0.000815 + 0.00146i)14-s + 0.905·15-s + (−0.499 − 0.866i)16-s + (−0.471 + 0.816i)17-s + (−0.00184 + 0.00318i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.902 + 0.431i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.902 + 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.174224692\)
\(L(\frac12)\) \(\approx\) \(2.174224692\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.64 + 0.0383i)T \)
13 \( 1 \)
good2 \( 1 + (0.00118 + 0.00205i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.54 + 2.68i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.565 - 0.979i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.23 + 3.87i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.94 - 3.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.27 - 2.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.459 - 0.796i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.32T + 29T^{2} \)
31 \( 1 + (3.12 - 5.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.769 - 1.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.86T + 41T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + (4.71 + 8.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.91 + 8.50i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.87 - 8.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.21 - 3.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.44 + 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + (-6.53 + 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0197 + 0.0342i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.33T + 83T^{2} \)
89 \( 1 + (-1.83 - 3.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.12T + 97T^{2} \)
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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

−9.170179449738901997681382627988, −8.702607874232726275320193356403, −7.55027197568447934120668644605, −6.85059659585082968955318393594, −6.21030438996862032313036865841, −5.79795344942637166252553150682, −3.68797530300745632437919567222, −2.84615397799948648577204659570, −1.91301026191852800913886414668, −0.811604755713985767507097402966, 2.29749841157763536997257406704, 3.05366808498089254534224420411, 4.03198578502505398066173952959, 4.55588828469166065177034096996, 5.79237125715652970385064083256, 7.05301760271161118393187971848, 7.67284477789376010689184247971, 8.946244061510756272096621826503, 9.289405675155825974485130174959, 9.709692874828528798960046156062

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