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Properties

Label 1-1157-1157.1101-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.200 - 0.979i$
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  + (0.723 + 0.690i)2-s + (−0.672 − 0.739i)3-s + (0.0475 + 0.998i)4-s + (−0.755 − 0.654i)5-s + (0.0237 − 0.999i)6-s + (−0.899 − 0.436i)7-s + (−0.654 + 0.755i)8-s + (−0.0950 + 0.995i)9-s + (−0.0950 − 0.995i)10-s + (0.327 − 0.945i)11-s + (0.707 − 0.707i)12-s + (−0.349 − 0.936i)14-s + (0.0237 + 0.999i)15-s + (−0.995 + 0.0950i)16-s + (0.690 + 0.723i)17-s + (−0.755 + 0.654i)18-s + ⋯
L(s)  = 1  + (0.723 + 0.690i)2-s + (−0.672 − 0.739i)3-s + (0.0475 + 0.998i)4-s + (−0.755 − 0.654i)5-s + (0.0237 − 0.999i)6-s + (−0.899 − 0.436i)7-s + (−0.654 + 0.755i)8-s + (−0.0950 + 0.995i)9-s + (−0.0950 − 0.995i)10-s + (0.327 − 0.945i)11-s + (0.707 − 0.707i)12-s + (−0.349 − 0.936i)14-s + (0.0237 + 0.999i)15-s + (−0.995 + 0.0950i)16-s + (0.690 + 0.723i)17-s + (−0.755 + 0.654i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.200 - 0.979i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.200 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5942816959 - 0.7285122318i\)
\(L(\frac12)\) \(\approx\) \(0.5942816959 - 0.7285122318i\)
\(L(1)\) \(\approx\) \(0.9256440944 + 0.02736230394i\)
\(L(1)\) \(\approx\) \(0.9256440944 + 0.02736230394i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.723 + 0.690i)T \)
3 \( 1 + (-0.672 - 0.739i)T \)
5 \( 1 + (-0.755 - 0.654i)T \)
7 \( 1 + (-0.899 - 0.436i)T \)
11 \( 1 + (0.327 - 0.945i)T \)
17 \( 1 + (0.690 + 0.723i)T \)
19 \( 1 + (0.636 + 0.771i)T \)
23 \( 1 + (-0.165 - 0.986i)T \)
29 \( 1 + (-0.560 - 0.828i)T \)
31 \( 1 + (0.936 - 0.349i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (0.952 - 0.304i)T \)
43 \( 1 + (-0.560 + 0.828i)T \)
47 \( 1 + (0.540 + 0.841i)T \)
53 \( 1 + (-0.540 + 0.841i)T \)
59 \( 1 + (0.672 - 0.739i)T \)
61 \( 1 + (0.992 - 0.118i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.189 + 0.981i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (-0.909 - 0.415i)T \)
83 \( 1 + (0.877 - 0.479i)T \)
97 \( 1 + (-0.327 - 0.945i)T \)
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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

−21.521409634250299821713129488347, −20.57972959033927126163276402200, −19.85580696063371091043610360595, −19.22453017803785349157348173920, −18.29807417048801119932033478410, −17.65000264314803988023363257503, −16.220198242845943748140260688705, −15.81787184753471743791081290517, −15.080662425399711242100320830293, −14.47957224631219705873882778193, −13.39126172737923879762497131019, −12.297068409873788980576518715944, −11.932517065352322764400486605622, −11.23967432804394830340720820595, −10.31266276008278586211035082228, −9.70382775867207349899387423636, −9.034577524388132120099453171073, −7.23309825766514147642823148552, −6.709305047177267360380461896559, −5.62678024341662515227645055672, −4.961541223193754262821693528790, −3.89137443995066953932705133046, −3.3671176134500775669113829227, −2.464288266847555965087713841797, −0.86856596438601765656603556274, 0.222502828464209121814300252120, 1.191491636911356288685110559527, 2.85004710997281688339116474035, 3.77383922673156193811036130031, 4.53740418160316552208748087613, 5.71648235665953482801606148354, 6.15021054934374587376359422353, 7.05523745720519140973994064004, 7.96507109963953629713250763316, 8.36731223289700892400791887933, 9.70651351027002691684791714180, 10.94831018623256974510879073627, 11.772560293466888650805841208010, 12.39513838341551093230003734242, 12.99008758008384235931948130971, 13.72063964172703934001109170430, 14.50828748426805707934284332611, 15.74696930789827778122102106440, 16.258734634441729513623962067617, 16.8623169317074029710325038284, 17.33319547763296782317748274509, 18.778394180147807334908289584979, 19.087204612405023451812011160142, 20.16051686780774299609086799696, 20.92454991714513320578380871748

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