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Properties

Label 1-1183-1183.349-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.342 - 0.939i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.534 − 0.845i)2-s + (−0.692 + 0.721i)3-s + (−0.428 + 0.903i)4-s + (−0.935 + 0.354i)5-s + (0.979 + 0.200i)6-s + (0.992 − 0.120i)8-s + (−0.0402 − 0.999i)9-s + (0.799 + 0.600i)10-s + (0.999 + 0.0402i)11-s + (−0.354 − 0.935i)12-s + (0.391 − 0.919i)15-s + (−0.632 − 0.774i)16-s + (0.799 − 0.600i)17-s + (−0.822 + 0.568i)18-s + (−0.866 + 0.5i)19-s + (0.0804 − 0.996i)20-s + ⋯
L(s)  = 1  + (−0.534 − 0.845i)2-s + (−0.692 + 0.721i)3-s + (−0.428 + 0.903i)4-s + (−0.935 + 0.354i)5-s + (0.979 + 0.200i)6-s + (0.992 − 0.120i)8-s + (−0.0402 − 0.999i)9-s + (0.799 + 0.600i)10-s + (0.999 + 0.0402i)11-s + (−0.354 − 0.935i)12-s + (0.391 − 0.919i)15-s + (−0.632 − 0.774i)16-s + (0.799 − 0.600i)17-s + (−0.822 + 0.568i)18-s + (−0.866 + 0.5i)19-s + (0.0804 − 0.996i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2173597550 - 0.3107633120i\)
\(L(\frac12)\) \(\approx\) \(0.2173597550 - 0.3107633120i\)
\(L(1)\) \(\approx\) \(0.5030757100 - 0.08537368198i\)
\(L(1)\) \(\approx\) \(0.5030757100 - 0.08537368198i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.534 - 0.845i)T \)
3 \( 1 + (-0.692 + 0.721i)T \)
5 \( 1 + (-0.935 + 0.354i)T \)
11 \( 1 + (0.999 + 0.0402i)T \)
17 \( 1 + (0.799 - 0.600i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.845 + 0.534i)T \)
31 \( 1 + (-0.663 + 0.748i)T \)
37 \( 1 + (-0.316 - 0.948i)T \)
41 \( 1 + (-0.721 - 0.692i)T \)
43 \( 1 + (-0.948 - 0.316i)T \)
47 \( 1 + (0.822 + 0.568i)T \)
53 \( 1 + (0.120 + 0.992i)T \)
59 \( 1 + (-0.774 - 0.632i)T \)
61 \( 1 + (0.919 - 0.391i)T \)
67 \( 1 + (-0.903 + 0.428i)T \)
71 \( 1 + (-0.960 + 0.278i)T \)
73 \( 1 + (0.464 + 0.885i)T \)
79 \( 1 + (0.568 - 0.822i)T \)
83 \( 1 + (0.239 + 0.970i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.160 - 0.987i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.73069157401049270760227508450, −20.353722508547755650674263490281, −19.42747848082653499283436787105, −19.17532889252791436375756790634, −18.381617619804535153663294263844, −17.37711343225700029111564811454, −16.798439180915001487632170513334, −16.46119122742098709898601471092, −15.169714520061222097315914356883, −14.89381116559303665551567987949, −13.59081149123309445349806830273, −12.983068079005265109550644449813, −11.90969907810575228358365895713, −11.35865727422242848420069213898, −10.461204546671590060518200997681, −9.36857819055846074837667090442, −8.487565617680493603172445693361, −7.80179357921401474242229619134, −7.06622639932497514764462060477, −6.34231399712137117331801974753, −5.46781274910672998144539988193, −4.60774148707851951884151298427, −3.63598583311432271147211061939, −1.80325783652313978266063856439, −0.9795540785168395068646354884, 0.26747421523340689086492012415, 1.49651155051566478183969293753, 2.983074367273482826602547596341, 3.753253201985786185632470114804, 4.32115887309243161113142230267, 5.36566174593260245827710179917, 6.69791695499148888722340297426, 7.37042666257166108852284640866, 8.58796017033928782769362274640, 9.13849386607904865114054873912, 10.16435842174003674247576845471, 10.8011189030533567952413234663, 11.40496878777468674744005822849, 12.25023509935883233885124138865, 12.56015797294263295288906866015, 14.16154101952270587619327466167, 14.783121077054180059053373349767, 15.79475219115106210134871942796, 16.67578105575038982925034223482, 16.9419242109412586217411696780, 18.06882242895306921949234326099, 18.735980962992976939158258417521, 19.40824665274401871297289170750, 20.36254997535533035966685341178, 20.782709401763853239470067914573

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