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Properties

Label 2-1183-7.4-c1-0-14
Degree $2$
Conductor $1183$
Sign $0.961 + 0.275i$
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  + (−0.973 − 1.68i)2-s + (0.867 − 1.50i)3-s + (−0.895 + 1.55i)4-s + (1.85 + 3.22i)5-s − 3.37·6-s + (−1.95 − 1.78i)7-s − 0.406·8-s + (−0.00432 − 0.00749i)9-s + (3.62 − 6.27i)10-s + (−1.78 + 3.08i)11-s + (1.55 + 2.69i)12-s + (−1.10 + 5.03i)14-s + 6.45·15-s + (2.18 + 3.78i)16-s + (−0.847 + 1.46i)17-s + (−0.00843 + 0.0146i)18-s + ⋯
L(s)  = 1  + (−0.688 − 1.19i)2-s + (0.500 − 0.867i)3-s + (−0.447 + 0.775i)4-s + (0.831 + 1.44i)5-s − 1.37·6-s + (−0.738 − 0.674i)7-s − 0.143·8-s + (−0.00144 − 0.00249i)9-s + (1.14 − 1.98i)10-s + (−0.537 + 0.930i)11-s + (0.448 + 0.776i)12-s + (−0.295 + 1.34i)14-s + 1.66·15-s + (0.546 + 0.946i)16-s + (−0.205 + 0.355i)17-s + (−0.00198 + 0.00344i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.961 + 0.275i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.961 + 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.123724460\)
\(L(\frac12)\) \(\approx\) \(1.123724460\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.95 + 1.78i)T \)
13 \( 1 \)
good2 \( 1 + (0.973 + 1.68i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.867 + 1.50i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.85 - 3.22i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.78 - 3.08i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.847 - 1.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.74 - 4.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.09 + 5.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.39T + 29T^{2} \)
31 \( 1 + (3.52 - 6.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.907 - 1.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.877T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + (-1.36 - 2.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.02 - 3.50i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.60 - 2.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.24 - 12.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.16 + 7.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.11T + 71T^{2} \)
73 \( 1 + (1.53 - 2.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.843 - 1.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.44T + 83T^{2} \)
89 \( 1 + (-2.23 - 3.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−9.963130845598054724422694606757, −9.278540245418837024878832316681, −8.077293165201952357856495316253, −7.31327865825662444880323237891, −6.65506795435424754345209483675, −5.81138632097019165595071435002, −3.97966064799491920284531930738, −2.88606772787558575822598641894, −2.35674551265425913062095449969, −1.43491848316388339308174139971, 0.57452198933167282670645158353, 2.55569200529998974119523103495, 3.72771362207797299369798864729, 5.10385480943403397328141810942, 5.59484659519638110801369650334, 6.34227910501886076277759023208, 7.54035419193328323749682125392, 8.452268500730529905287268743880, 9.045441975455733116386351674401, 9.478829668757018978043090032107

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