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Properties

Label 1-1157-1157.1046-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.856 - 0.516i$
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  + (−0.814 − 0.580i)2-s + (−0.327 + 0.945i)3-s + (0.327 + 0.945i)4-s + (−0.281 + 0.959i)5-s + (0.814 − 0.580i)6-s + (0.690 + 0.723i)7-s + (0.281 − 0.959i)8-s + (−0.786 − 0.618i)9-s + (0.786 − 0.618i)10-s + (−0.690 + 0.723i)11-s − 12-s + (−0.142 − 0.989i)14-s + (−0.814 − 0.580i)15-s + (−0.786 + 0.618i)16-s + (−0.580 − 0.814i)17-s + (0.281 + 0.959i)18-s + ⋯
L(s)  = 1  + (−0.814 − 0.580i)2-s + (−0.327 + 0.945i)3-s + (0.327 + 0.945i)4-s + (−0.281 + 0.959i)5-s + (0.814 − 0.580i)6-s + (0.690 + 0.723i)7-s + (0.281 − 0.959i)8-s + (−0.786 − 0.618i)9-s + (0.786 − 0.618i)10-s + (−0.690 + 0.723i)11-s − 12-s + (−0.142 − 0.989i)14-s + (−0.814 − 0.580i)15-s + (−0.786 + 0.618i)16-s + (−0.580 − 0.814i)17-s + (0.281 + 0.959i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.856 - 0.516i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1046, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.856 - 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1559970348 - 0.04342408161i\)
\(L(\frac12)\) \(\approx\) \(0.1559970348 - 0.04342408161i\)
\(L(1)\) \(\approx\) \(0.4643214763 + 0.2116411777i\)
\(L(1)\) \(\approx\) \(0.4643214763 + 0.2116411777i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.814 - 0.580i)T \)
3 \( 1 + (-0.327 + 0.945i)T \)
5 \( 1 + (-0.281 + 0.959i)T \)
7 \( 1 + (0.690 + 0.723i)T \)
11 \( 1 + (-0.690 + 0.723i)T \)
17 \( 1 + (-0.580 - 0.814i)T \)
19 \( 1 + (-0.618 + 0.786i)T \)
23 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (0.235 + 0.971i)T \)
31 \( 1 + (-0.989 + 0.142i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.189 - 0.981i)T \)
43 \( 1 + (-0.235 + 0.971i)T \)
47 \( 1 + (-0.755 - 0.654i)T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (0.945 - 0.327i)T \)
61 \( 1 + (0.0475 + 0.998i)T \)
67 \( 1 + (0.945 + 0.327i)T \)
71 \( 1 + (-0.971 - 0.235i)T \)
73 \( 1 + (0.989 - 0.142i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (-0.909 - 0.415i)T \)
97 \( 1 + (-0.690 - 0.723i)T \)
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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

−20.86508924180059564885825527040, −20.00791458096387151297440811628, −19.61849996987892232559455091356, −18.69751261630073684717389551588, −17.959550297309323676117935881477, −17.23859813008694438424453329985, −16.789980751798273471169317415502, −15.99848274452771819775652826010, −15.09796880877970466037185052405, −14.06576211290124492175022909214, −13.36605207273637950598839346364, −12.654918012901424749052487454666, −11.359951722438887671161959341887, −11.10585034986095373657412850589, −10.04623527845112446169546320001, −8.82232042696970937619051693346, −8.099145734398266069378572999095, −7.864401381264650577555024176925, −6.73078850847342057516456656502, −5.9523428934842688979062667365, −5.09023569435113781954528979820, −4.22569352347938316720047216399, −2.38840670817951021024647835677, −1.50508223282864614412751244527, −0.59800034806865698015668135594, 0.0722583271273316363940553739, 1.89943633139523854880761575687, 2.62397808840096513847907384077, 3.59741889355619124853046269760, 4.4740714610443643151888307808, 5.51094108891965432603022618995, 6.61705528852192080597824042337, 7.593160183909926940799120775683, 8.36766534602493741607492014271, 9.28116782498072865837601551784, 10.061196731224622409302173137725, 10.70364524952972560414993456713, 11.38644765097601459228040572588, 11.95695659743925972890375863139, 12.88445954060778145872520751592, 14.347262395547137510051529415424, 14.90852507330892041688601995806, 15.82891897381438908364524530943, 16.22444129162594246513279962733, 17.48204144301263626645061791568, 18.077925108059911666042885955598, 18.39709845127740213816331305711, 19.556696775265742075551748167436, 20.29906009067825763562923733323, 21.01901695319080619349758964514

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