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Properties

Label 1-1157-1157.128-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.947 - 0.319i$
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  + (0.945 − 0.327i)2-s + (−0.786 − 0.618i)3-s + (0.786 − 0.618i)4-s + (−0.540 + 0.841i)5-s + (−0.945 − 0.327i)6-s + (−0.998 − 0.0475i)7-s + (0.540 − 0.841i)8-s + (0.235 + 0.971i)9-s + (−0.235 + 0.971i)10-s + (0.998 − 0.0475i)11-s − 12-s + (−0.959 + 0.281i)14-s + (0.945 − 0.327i)15-s + (0.235 − 0.971i)16-s + (0.327 − 0.945i)17-s + (0.540 + 0.841i)18-s + ⋯
L(s)  = 1  + (0.945 − 0.327i)2-s + (−0.786 − 0.618i)3-s + (0.786 − 0.618i)4-s + (−0.540 + 0.841i)5-s + (−0.945 − 0.327i)6-s + (−0.998 − 0.0475i)7-s + (0.540 − 0.841i)8-s + (0.235 + 0.971i)9-s + (−0.235 + 0.971i)10-s + (0.998 − 0.0475i)11-s − 12-s + (−0.959 + 0.281i)14-s + (0.945 − 0.327i)15-s + (0.235 − 0.971i)16-s + (0.327 − 0.945i)17-s + (0.540 + 0.841i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.963371360 - 0.3223082079i\)
\(L(\frac12)\) \(\approx\) \(1.963371360 - 0.3223082079i\)
\(L(1)\) \(\approx\) \(1.164776655 - 0.3196616139i\)
\(L(1)\) \(\approx\) \(1.164776655 - 0.3196616139i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.945 - 0.327i)T \)
3 \( 1 + (-0.786 - 0.618i)T \)
5 \( 1 + (-0.540 + 0.841i)T \)
7 \( 1 + (-0.998 - 0.0475i)T \)
11 \( 1 + (0.998 - 0.0475i)T \)
17 \( 1 + (0.327 - 0.945i)T \)
19 \( 1 + (-0.971 + 0.235i)T \)
23 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (-0.888 + 0.458i)T \)
31 \( 1 + (-0.281 - 0.959i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.371 + 0.928i)T \)
43 \( 1 + (0.888 + 0.458i)T \)
47 \( 1 + (-0.989 + 0.142i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (0.618 + 0.786i)T \)
61 \( 1 + (-0.995 + 0.0950i)T \)
67 \( 1 + (0.618 - 0.786i)T \)
71 \( 1 + (0.458 - 0.888i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (0.755 - 0.654i)T \)
97 \( 1 + (0.998 + 0.0475i)T \)
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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

−21.36902440309628238327631125039, −20.521211835190336691982142389077, −19.787858686492743351528758854409, −19.09091064969311007631796156670, −17.567273260567628518779228398202, −16.86828020054251010162435569941, −16.45581564865503632423768339584, −15.75820100835200516632857889046, −15.0564207241740690232776158614, −14.28887495950366043290975985891, −12.91753024960125049595553102484, −12.62653938420619117084671958937, −11.905294777275176842137982862252, −11.11319059005067389247024060697, −10.18699239505246183370642366458, −9.15495680091764801859026145113, −8.36753335057149998530739980319, −7.11471894340708354901375606640, −6.311332588115813230559161912272, −5.741755636559975008069444753285, −4.71625236494363545726260635364, −3.909044181656968520740956779319, −3.57066669422939331029598715580, −1.92210861720143201475853701180, −0.48042427667264211452974318366, 0.657567907161504062839464655349, 1.88264391498953266374589610202, 2.902628796117017424087607146, 3.76680940788654855431890623437, 4.58600762603176593112845727238, 5.9598919257037936013950284845, 6.25787769669938404562983206691, 7.10908687727940136318701928891, 7.742912163341395948725886708309, 9.46674348407351908958270122073, 10.2225646223089255281519468343, 11.20394385988171334107753109102, 11.61304540461562282907420659887, 12.39450145041046125169754334253, 13.113573565376767985524350443890, 13.91608909149248155436026596904, 14.69367146542530601190321793363, 15.55316398713754056506027918712, 16.39326491415651306393149858601, 16.93140521845055212750064797782, 18.32734728819484802381015638190, 18.78547540620490316093955617147, 19.631465930018865358784147638129, 19.98471627686482483334447408423, 21.38214238033595989792325490333

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