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Properties

Label 1-1157-1157.212-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.473 + 0.880i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + 5-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s − 8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + i·12-s + i·14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + 5-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s − 8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + i·12-s + i·14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.473 + 0.880i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.473 + 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.006313462 + 1.797012864i\)
\(L(\frac12)\) \(\approx\) \(3.006313462 + 1.797012864i\)
\(L(1)\) \(\approx\) \(2.020915028 + 0.8392928854i\)
\(L(1)\) \(\approx\) \(2.020915028 + 0.8392928854i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - iT \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−21.08210829719876901471958694472, −20.65994926643699185407487789017, −19.80751742665940586649887710069, −18.95468728317057467399036500474, −18.45401470977630731572189796027, −17.08745751014509125602880760855, −16.82942284833074437978470253984, −15.14758584055815713558584928866, −14.78969490371995036383602122669, −13.9948142116698606589975912602, −13.47835806528482747176187690086, −12.811941784627838001586055190351, −11.52659303203835040758474268172, −10.769012374959581351498998016639, −10.20849136229113637412774745302, −9.34208802808447999880148734513, −8.66886174801258670626161912642, −7.77016727566850263283365815614, −6.303301524354684322974437066294, −5.52360307336729454342022317066, −4.561242435826100494682447121624, −3.81393404691154179021313047652, −2.95706645275776181076524416616, −1.87840655890162169094479744639, −1.33033788234401513383548210395, 1.415157239432724298097805076288, 2.34284730005230224013273654476, 3.16405766868519674808278338641, 4.49480737584578429028475956020, 5.11042343152220021607235744057, 6.1860900083749294182720840324, 6.97250758163501060731918702351, 7.59961225016389554014713121244, 8.827769878433938556977120759024, 8.9722849841646400311942385986, 10.02864858133360506466134618845, 11.436248681485076544068836753169, 12.45696530428915577946880660800, 12.925131441273761302930369917778, 13.83001167711050209681786199938, 14.56057444526198334296539585806, 14.80072251956650140693805281058, 15.734780554798208558597255061574, 16.99099810588691691972823697476, 17.44266243850185236080450301825, 18.34049769596452550832955834574, 18.67977144399673841148843144583, 20.19072196973260518040514802226, 20.7023797790496195924753239847, 21.51538985732265486191616600230

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