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Properties

Label 1-1157-1157.18-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.168 + 0.985i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.168 + 0.985i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.168 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.099435982 + 4.857384248i\)
\(L(\frac12)\) \(\approx\) \(4.099435982 + 4.857384248i\)
\(L(1)\) \(\approx\) \(2.352874518 + 1.252624043i\)
\(L(1)\) \(\approx\) \(2.352874518 + 1.252624043i\)

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

−20.96083057905523958750204549431, −20.391224819302326777384279695305, −19.307157466805114712480390596438, −18.67968745253821373988977157406, −17.88816356467944703028218043562, −17.06492921844919745491810637777, −16.08535351360770963444528933586, −15.16682279977570311795238246285, −14.44058394497568482887639410088, −13.676494161362591550104870510392, −13.13963662967316843016576354627, −12.3744095130794007319649235344, −11.85078062158419597929591401354, −10.770164393491170987258178692276, −9.900820986771044692285305136685, −8.60341340443838521493363940990, −8.14173208089181879298277324932, −6.94025565912504044192312010378, −6.02717190150422708241622488832, −5.58638558961154020151996706571, −4.737719179887942979907312952029, −3.283569359276451806791931518741, −2.42968987289618656039018955825, −1.86886764276771970447073947222, −0.817810912265084986314988290977, 1.28683869733078396318682194618, 2.648009570514751464162964128659, 3.03164119818768324017510573356, 4.23488242211389873684904134265, 5.07931042454454066897008169725, 5.43333941869062785001263722269, 6.82883181419310229349253289221, 7.38636113764433053483422163151, 8.4832564407966895963759304190, 9.74342964076314830502576136707, 10.269132717006626489917702267180, 11.036153705627130653817541804648, 11.71874988466370447158081031072, 13.149038906204221708852465431127, 13.63807587446943230651580920651, 14.243345101503922390912087503325, 14.92222477765010302721184011067, 15.86358523445763619713109012201, 16.289491806099477550835362845626, 17.50567419701119021286620387299, 17.762092515067135123054571530993, 19.32194661438548886840578829810, 20.19274396566979424703438328843, 20.80741421224836309434253703493, 21.24911571476715779577275007110

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