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Properties

Label 1-1183-1183.346-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.997 - 0.0714i$
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  + (−0.600 + 0.799i)2-s + (−0.948 − 0.316i)3-s + (−0.278 − 0.960i)4-s + (0.999 + 0.0402i)5-s + (0.822 − 0.568i)6-s + (0.935 + 0.354i)8-s + (0.799 + 0.600i)9-s + (−0.632 + 0.774i)10-s + (−0.600 − 0.799i)11-s + (−0.0402 + 0.999i)12-s + (−0.935 − 0.354i)15-s + (−0.845 + 0.534i)16-s + (0.987 − 0.160i)17-s + (−0.960 + 0.278i)18-s + (0.866 + 0.5i)19-s + (−0.239 − 0.970i)20-s + ⋯
L(s)  = 1  + (−0.600 + 0.799i)2-s + (−0.948 − 0.316i)3-s + (−0.278 − 0.960i)4-s + (0.999 + 0.0402i)5-s + (0.822 − 0.568i)6-s + (0.935 + 0.354i)8-s + (0.799 + 0.600i)9-s + (−0.632 + 0.774i)10-s + (−0.600 − 0.799i)11-s + (−0.0402 + 0.999i)12-s + (−0.935 − 0.354i)15-s + (−0.845 + 0.534i)16-s + (0.987 − 0.160i)17-s + (−0.960 + 0.278i)18-s + (0.866 + 0.5i)19-s + (−0.239 − 0.970i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.997 - 0.0714i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.997 - 0.0714i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9683336203 - 0.03464990428i\)
\(L(\frac12)\) \(\approx\) \(0.9683336203 - 0.03464990428i\)
\(L(1)\) \(\approx\) \(0.7408810713 + 0.09151789581i\)
\(L(1)\) \(\approx\) \(0.7408810713 + 0.09151789581i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.600 + 0.799i)T \)
3 \( 1 + (-0.948 - 0.316i)T \)
5 \( 1 + (0.999 + 0.0402i)T \)
11 \( 1 + (-0.600 - 0.799i)T \)
17 \( 1 + (0.987 - 0.160i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.120 - 0.992i)T \)
31 \( 1 + (-0.903 - 0.428i)T \)
37 \( 1 + (0.0804 + 0.996i)T \)
41 \( 1 + (0.663 + 0.748i)T \)
43 \( 1 + (-0.568 + 0.822i)T \)
47 \( 1 + (0.960 + 0.278i)T \)
53 \( 1 + (0.987 - 0.160i)T \)
59 \( 1 + (0.534 - 0.845i)T \)
61 \( 1 + (-0.987 - 0.160i)T \)
67 \( 1 + (-0.721 + 0.692i)T \)
71 \( 1 + (-0.663 - 0.748i)T \)
73 \( 1 + (0.600 + 0.799i)T \)
79 \( 1 + (0.278 - 0.960i)T \)
83 \( 1 + (-0.663 + 0.748i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.464 - 0.885i)T \)
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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

−21.29647417366414434723661604203, −20.60940727630808890403097977020, −19.803831812360060511726689150831, −18.62611405761579182387464476997, −18.05627987899442747971939017089, −17.60074745399180620758044915929, −16.807257234372983794013828878063, −16.19869260112475412752845788444, −15.2005918855534472359614967393, −14.02719808237578507209099159628, −13.11664213289830664428751366516, −12.467352629220703470683890502864, −11.80855947342025275424653813167, −10.65342875473949081645514102743, −10.42430525492857740922831447028, −9.461170057966396681236288293357, −9.02125537711820505506185315036, −7.47276708726384949843570096909, −7.01050367291479805058753918840, −5.52473538708954723628139721433, −5.17814064813165656916823341665, −3.96521154722506824590912426466, −2.93610481743925851714389036667, −1.81379145467664097505441551714, −0.97534129064059676601694023762, 0.727199432964252439061592722632, 1.56669688878665519724073536585, 2.81969630215950854686736547152, 4.52397157495476078613936576194, 5.50280024491422353107914191783, 5.80621387959725849111635712607, 6.64067943500940811559176077825, 7.556737530103290170382136204421, 8.29351434343266443433375141468, 9.475920203326833272101971690615, 10.083349688044431923102703025, 10.77545025055079247000177965592, 11.62593221341974927325772207822, 12.80658307624114010392080890418, 13.52052128361415706787133932891, 14.20349362051365177880103275099, 15.1472485547329264839417879314, 16.28840542229062469963404704819, 16.57655194506397348745044898688, 17.277154543004666558844301367463, 18.245652149694573027989055916861, 18.46617245981711910703994730291, 19.21290174371601906293536326363, 20.5182110301453523606199522660, 21.28400567305080961711163016776

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