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Properties

Label 1-1169-1169.2-r0-0-0
Degree $1$
Conductor $1169$
Sign $0.928 + 0.372i$
Analytic cond. $5.42881$
Root an. cond. $5.42881$
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  + (−0.340 + 0.940i)2-s + (0.994 − 0.100i)3-s + (−0.768 − 0.639i)4-s + (0.836 − 0.548i)5-s + (−0.243 + 0.969i)6-s + (0.862 − 0.505i)8-s + (0.979 − 0.200i)9-s + (0.231 + 0.972i)10-s + (0.875 + 0.483i)11-s + (−0.829 − 0.558i)12-s + (0.421 − 0.906i)13-s + (0.776 − 0.629i)15-s + (0.181 + 0.983i)16-s + (0.792 + 0.610i)17-s + (−0.144 + 0.989i)18-s + (−0.624 − 0.780i)19-s + ⋯
L(s)  = 1  + (−0.340 + 0.940i)2-s + (0.994 − 0.100i)3-s + (−0.768 − 0.639i)4-s + (0.836 − 0.548i)5-s + (−0.243 + 0.969i)6-s + (0.862 − 0.505i)8-s + (0.979 − 0.200i)9-s + (0.231 + 0.972i)10-s + (0.875 + 0.483i)11-s + (−0.829 − 0.558i)12-s + (0.421 − 0.906i)13-s + (0.776 − 0.629i)15-s + (0.181 + 0.983i)16-s + (0.792 + 0.610i)17-s + (−0.144 + 0.989i)18-s + (−0.624 − 0.780i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1169\)    =    \(7 \cdot 167\)
Sign: $0.928 + 0.372i$
Analytic conductor: \(5.42881\)
Root analytic conductor: \(5.42881\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1169} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1169,\ (0:\ ),\ 0.928 + 0.372i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.246263067 + 0.4334879728i\)
\(L(\frac12)\) \(\approx\) \(2.246263067 + 0.4334879728i\)
\(L(1)\) \(\approx\) \(1.457137979 + 0.3295389360i\)
\(L(1)\) \(\approx\) \(1.457137979 + 0.3295389360i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
167 \( 1 \)
good2 \( 1 + (-0.340 + 0.940i)T \)
3 \( 1 + (0.994 - 0.100i)T \)
5 \( 1 + (0.836 - 0.548i)T \)
11 \( 1 + (0.875 + 0.483i)T \)
13 \( 1 + (0.421 - 0.906i)T \)
17 \( 1 + (0.792 + 0.610i)T \)
19 \( 1 + (-0.624 - 0.780i)T \)
23 \( 1 + (-0.990 - 0.138i)T \)
29 \( 1 + (0.614 + 0.788i)T \)
31 \( 1 + (0.992 + 0.125i)T \)
37 \( 1 + (-0.663 + 0.748i)T \)
41 \( 1 + (-0.700 + 0.713i)T \)
43 \( 1 + (0.974 - 0.225i)T \)
47 \( 1 + (-0.986 + 0.163i)T \)
53 \( 1 + (-0.814 - 0.579i)T \)
59 \( 1 + (-0.924 - 0.381i)T \)
61 \( 1 + (-0.904 - 0.427i)T \)
67 \( 1 + (0.836 + 0.548i)T \)
71 \( 1 + (0.553 - 0.832i)T \)
73 \( 1 + (0.181 - 0.983i)T \)
79 \( 1 + (-0.477 + 0.878i)T \)
83 \( 1 + (-0.644 + 0.764i)T \)
89 \( 1 + (0.157 - 0.987i)T \)
97 \( 1 + (-0.387 + 0.922i)T \)
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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

−21.27976080712820171585318126477, −20.530157108447403162641053446326, −19.56583003215959124494021786674, −18.92018093876059865466422652988, −18.54767563291954074796103444565, −17.51936071529867039975546679475, −16.75629213682361023164404247434, −15.858380275020328750454202403770, −14.54316869149653281109385491526, −13.890090602261526415880625390617, −13.79582779381299853961545810974, −12.5244458985041063247152660075, −11.78119810057869120366452897181, −10.783781038475857102731659421456, −9.96960994721361495915148866341, −9.47171673079453851000737875292, −8.68960311788756715659613500220, −7.91520828330895047154646555768, −6.85052781552254512341209357460, −5.87667427687978754948609116209, −4.44549920190343638885526859184, −3.71593225195316550412730213900, −2.90357080178551985325506235885, −1.98870276826717775347889867832, −1.32167228853067841715414934644, 1.10731197920127469026981774370, 1.80593660028251186691958444615, 3.18479577511449847754592490362, 4.30481408246329400204156143781, 5.06853236113628570594420195819, 6.24708446253800433375547139694, 6.719792831555390060680085679796, 8.02278962335972790201768133951, 8.38437778554737048422841109947, 9.26425014391598930103756628368, 9.905432149945959713386660906730, 10.55458270467889116259872651667, 12.3304872385568903574521083594, 12.92019033853830385316213869168, 13.806605575943736712097331457481, 14.28227862011296913352856653316, 15.12003328925440003866833298770, 15.772185182273270878195635634828, 16.71579514812419676195976224234, 17.49504889592631857333156535905, 18.00316749875666295921143291477, 18.95776992968379584639534618654, 19.76648953745489593205742059224, 20.30567134693561880030045253618, 21.31362276214788484676048161804

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