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Properties

Label 1-1157-1157.5-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.406 - 0.913i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (−0.989 + 0.142i)2-s + (0.281 − 0.959i)3-s + (0.959 − 0.281i)4-s + (−0.909 + 0.415i)5-s + (−0.142 + 0.989i)6-s + (−0.415 − 0.909i)7-s + (−0.909 + 0.415i)8-s + (−0.841 − 0.540i)9-s + (0.841 − 0.540i)10-s + (0.909 + 0.415i)11-s − i·12-s + (0.540 + 0.841i)14-s + (0.142 + 0.989i)15-s + (0.841 − 0.540i)16-s + (−0.142 + 0.989i)17-s + (0.909 + 0.415i)18-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (0.281 − 0.959i)3-s + (0.959 − 0.281i)4-s + (−0.909 + 0.415i)5-s + (−0.142 + 0.989i)6-s + (−0.415 − 0.909i)7-s + (−0.909 + 0.415i)8-s + (−0.841 − 0.540i)9-s + (0.841 − 0.540i)10-s + (0.909 + 0.415i)11-s − i·12-s + (0.540 + 0.841i)14-s + (0.142 + 0.989i)15-s + (0.841 − 0.540i)16-s + (−0.142 + 0.989i)17-s + (0.909 + 0.415i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.406 - 0.913i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.406 - 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7486025978 - 0.4859987432i\)
\(L(\frac12)\) \(\approx\) \(0.7486025978 - 0.4859987432i\)
\(L(1)\) \(\approx\) \(0.5944786517 - 0.1763472414i\)
\(L(1)\) \(\approx\) \(0.5944786517 - 0.1763472414i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.989 + 0.142i)T \)
3 \( 1 + (0.281 - 0.959i)T \)
5 \( 1 + (-0.909 + 0.415i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (0.909 + 0.415i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (0.540 - 0.841i)T \)
29 \( 1 + (0.909 - 0.415i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 + (0.281 + 0.959i)T \)
53 \( 1 + (0.959 + 0.281i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (0.755 - 0.654i)T \)
67 \( 1 + (-0.281 + 0.959i)T \)
71 \( 1 + (0.909 + 0.415i)T \)
73 \( 1 + (0.540 + 0.841i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.909 - 0.415i)T \)
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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

−21.21732697950781296302217574089, −20.0969486323785828513114145706, −19.83616479385896936963080076101, −19.01321117904566851016676537853, −18.36524842860233738994612374656, −17.105392412016141007965326233353, −16.50681119733996632302904411617, −16.00629713621609764271484163605, −15.17108880132127650960791150645, −14.7557789339583816410597735558, −13.33906564582505963304053932109, −12.15487316826525861067889126682, −11.65480649902695386211042821312, −10.97931333769200323087738189533, −9.924837299628564421153818704, −9.13230537803818514302705866590, −8.75177132644674360402160235406, −7.979532690139419183928559643638, −6.89819324711269693069453814583, −5.88451699675120119777096683809, −4.85165271402410211755243219484, −3.66577821310238317902818138014, −3.13091308813904170865982951246, −1.97384756619407122616051912728, −0.54470983488776901331411718059, 0.46723105094239864488667948647, 1.30929470311917145030344254469, 2.44146668619267149728990927112, 3.39948497139317557042627268621, 4.362924855437785608133782718017, 6.151565594695217557046844196581, 6.84926678693612656544281431535, 7.12562207057764818284126055873, 8.2307242051199333844021082354, 8.65454814624648900105249807054, 9.789712246295996707672889955, 10.73724484188326474656979872108, 11.3373586116337917364815354728, 12.28726775056613174297703365078, 12.892606112146639145129935766089, 14.15010103186346894245624173058, 14.81310092471599266722099444266, 15.46480366225661387360115796098, 16.69436997496542585005538610260, 17.07503798671762113379732389460, 17.931015351907045401922146850072, 18.792927374612623294004912330165, 19.376020057898232799790551021, 19.935193464748015136344759654809, 20.28474294626877173738043519240

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