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Properties

Label 1-1157-1157.167-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.455 - 0.890i$
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  + (0.618 − 0.786i)2-s + (0.235 − 0.971i)3-s + (−0.235 − 0.971i)4-s + (0.909 + 0.415i)5-s + (−0.618 − 0.786i)6-s + (0.0950 + 0.995i)7-s + (−0.909 − 0.415i)8-s + (−0.888 − 0.458i)9-s + (0.888 − 0.458i)10-s + (−0.0950 + 0.995i)11-s − 12-s + (0.841 + 0.540i)14-s + (0.618 − 0.786i)15-s + (−0.888 + 0.458i)16-s + (0.786 − 0.618i)17-s + (−0.909 + 0.415i)18-s + ⋯
L(s)  = 1  + (0.618 − 0.786i)2-s + (0.235 − 0.971i)3-s + (−0.235 − 0.971i)4-s + (0.909 + 0.415i)5-s + (−0.618 − 0.786i)6-s + (0.0950 + 0.995i)7-s + (−0.909 − 0.415i)8-s + (−0.888 − 0.458i)9-s + (0.888 − 0.458i)10-s + (−0.0950 + 0.995i)11-s − 12-s + (0.841 + 0.540i)14-s + (0.618 − 0.786i)15-s + (−0.888 + 0.458i)16-s + (0.786 − 0.618i)17-s + (−0.909 + 0.415i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.455 - 0.890i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.455 - 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.300858351 - 2.018598010i\)
\(L(\frac12)\) \(\approx\) \(3.300858351 - 2.018598010i\)
\(L(1)\) \(\approx\) \(1.546930950 - 0.9388138031i\)
\(L(1)\) \(\approx\) \(1.546930950 - 0.9388138031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.618 - 0.786i)T \)
3 \( 1 + (0.235 - 0.971i)T \)
5 \( 1 + (0.909 + 0.415i)T \)
7 \( 1 + (0.0950 + 0.995i)T \)
11 \( 1 + (-0.0950 + 0.995i)T \)
17 \( 1 + (0.786 - 0.618i)T \)
19 \( 1 + (0.458 - 0.888i)T \)
23 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (0.580 + 0.814i)T \)
31 \( 1 + (-0.540 + 0.841i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.690 + 0.723i)T \)
43 \( 1 + (-0.580 + 0.814i)T \)
47 \( 1 + (-0.281 + 0.959i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (0.971 - 0.235i)T \)
61 \( 1 + (0.981 + 0.189i)T \)
67 \( 1 + (0.971 + 0.235i)T \)
71 \( 1 + (0.814 + 0.580i)T \)
73 \( 1 + (0.540 - 0.841i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (0.989 - 0.142i)T \)
97 \( 1 + (-0.0950 - 0.995i)T \)
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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

−21.31336245783715540987018698800, −20.79444350495168825691351437334, −20.04676087975025622413673257812, −18.89202963781930937341909637939, −17.66472993788653220005798097986, −17.05552817305515967435719575522, −16.49302275141798703918478852456, −15.98370201847273078340370302172, −14.868817678942794010937971757814, −14.179355312951595861635550625773, −13.69617695463723152770780793006, −13.01884741967772987488366292152, −11.836679570223063609030575311083, −10.89177758127233688533619369920, −9.96538256859540777958949899685, −9.338516517240874405639629724, −8.24972060978867184894163450018, −7.7875618518930274699256231732, −6.41165261874972335854254490817, −5.61347448668928997654860187754, −5.13128370656793175082014054852, −3.856176287081192393146849655192, −3.593680810281650542167179213, −2.246314144543948109634244159877, −0.67811352354521450613135421286, 0.96919727897310990786349951509, 1.8395378873071916807279974774, 2.67004751011523261458301207409, 3.06136487056089312134608553895, 4.80329558907065687809451328659, 5.40926379553634288490804456506, 6.38496241193027659591095287075, 6.94732780649359126357785413673, 8.238651793950705450949136505772, 9.331197928250404124898704620032, 9.71802562345231981432099108570, 10.92425636434602157098246431192, 11.670899109813749186975886980559, 12.62317085332240848499390111375, 12.81887990175715072808822738229, 13.93065450524023054501881635410, 14.49633627280157581541308194953, 15.01575687665958341515863534086, 16.21488465535326325580284586892, 17.68985825549021842889498035732, 18.06826798321165739528666901417, 18.58671974871220188496074788733, 19.43302692078253231700914745664, 20.27296787047109268391668792028, 20.89317267195886355462088398716

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