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Properties

Label 1-1183-1183.79-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.323 - 0.946i$
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  + (0.428 − 0.903i)2-s + (−0.845 + 0.534i)3-s + (−0.632 − 0.774i)4-s + (0.948 + 0.316i)5-s + (0.120 + 0.992i)6-s + (−0.970 + 0.239i)8-s + (0.428 − 0.903i)9-s + (0.692 − 0.721i)10-s + (0.428 + 0.903i)11-s + (0.948 + 0.316i)12-s + (−0.970 + 0.239i)15-s + (−0.200 + 0.979i)16-s + (0.278 − 0.960i)17-s + (−0.632 − 0.774i)18-s + (−0.5 − 0.866i)19-s + (−0.354 − 0.935i)20-s + ⋯
L(s)  = 1  + (0.428 − 0.903i)2-s + (−0.845 + 0.534i)3-s + (−0.632 − 0.774i)4-s + (0.948 + 0.316i)5-s + (0.120 + 0.992i)6-s + (−0.970 + 0.239i)8-s + (0.428 − 0.903i)9-s + (0.692 − 0.721i)10-s + (0.428 + 0.903i)11-s + (0.948 + 0.316i)12-s + (−0.970 + 0.239i)15-s + (−0.200 + 0.979i)16-s + (0.278 − 0.960i)17-s + (−0.632 − 0.774i)18-s + (−0.5 − 0.866i)19-s + (−0.354 − 0.935i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.323 - 0.946i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.323 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.243811884 - 0.8888836716i\)
\(L(\frac12)\) \(\approx\) \(1.243811884 - 0.8888836716i\)
\(L(1)\) \(\approx\) \(1.050275770 - 0.4124409024i\)
\(L(1)\) \(\approx\) \(1.050275770 - 0.4124409024i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.428 - 0.903i)T \)
3 \( 1 + (-0.845 + 0.534i)T \)
5 \( 1 + (0.948 + 0.316i)T \)
11 \( 1 + (0.428 + 0.903i)T \)
17 \( 1 + (0.278 - 0.960i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.568 + 0.822i)T \)
31 \( 1 + (-0.919 - 0.391i)T \)
37 \( 1 + (0.799 - 0.600i)T \)
41 \( 1 + (0.885 + 0.464i)T \)
43 \( 1 + (0.120 - 0.992i)T \)
47 \( 1 + (-0.632 + 0.774i)T \)
53 \( 1 + (0.278 - 0.960i)T \)
59 \( 1 + (-0.200 - 0.979i)T \)
61 \( 1 + (0.278 + 0.960i)T \)
67 \( 1 + (0.987 + 0.160i)T \)
71 \( 1 + (0.885 + 0.464i)T \)
73 \( 1 + (0.428 + 0.903i)T \)
79 \( 1 + (-0.632 + 0.774i)T \)
83 \( 1 + (0.885 - 0.464i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.748 - 0.663i)T \)
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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

−21.613789124416770514792774271995, −21.11066868288138776886040503744, −19.668899197693688491934950230243, −18.77574502869598418295397404692, −18.00964936091190717851620587649, −17.38219203978606823268183614879, −16.66641917451830380022018088294, −16.36041547706744970834659870559, −15.19009140733302247122679274460, −14.18186156346935457337958116703, −13.66874693587016394510425386706, −12.85219640645879014884659608482, −12.305994003514510644347844312655, −11.3398250505530090052788162294, −10.312525643882835512620348294716, −9.40214256469744442039594496316, −8.36709534282608214203764151193, −7.73849465362880400409175827077, −6.5205922935995978790080185660, −6.036177416150885936364133469153, −5.55226357599981427729525100449, −4.53008494544535753469662921898, −3.53877334308652341819512247787, −2.105820577325100674219184179836, −1.00171379931734940888961639317, 0.74796469976635900001658076689, 1.94310445711539779927625021949, 2.79773250652529110700546856941, 3.98197329923332159162208745284, 4.77573913858288830748989766678, 5.44445592261480979763745987715, 6.35919195676460936938228898776, 7.049759320595187923437226909639, 8.854324886111134784430524907987, 9.59456234328768800649107668419, 10.04885280563690679017535275309, 10.94558499727634318295663564002, 11.47891197883281280100602611783, 12.565486842888499380776924430168, 12.91368737734973721844060063884, 14.21410200408292309580923150526, 14.57852572196169936407665707583, 15.554201051122518951978175348256, 16.55093733463974625916385023187, 17.47431871325063849286617111889, 18.027224009315206278164731166530, 18.56448590522001710819191474665, 19.80453920816098813682439981104, 20.49869342649268744963967484943, 21.1906023318452970022759522791

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