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Properties

Label 1-1157-1157.258-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.999 + 0.0109i$
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 + (0.618 − 0.786i)3-s + (0.786 − 0.618i)4-s + (0.540 − 0.841i)5-s + (0.327 − 0.945i)6-s + (0.0475 − 0.998i)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 − i·12-s + (−0.281 − 0.959i)14-s + (−0.327 − 0.945i)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 + (0.618 − 0.786i)3-s + (0.786 − 0.618i)4-s + (0.540 − 0.841i)5-s + (0.327 − 0.945i)6-s + (0.0475 − 0.998i)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 − i·12-s + (−0.281 − 0.959i)14-s + (−0.327 − 0.945i)15-s + (0.235 − 0.971i)16-s + (−0.327 + 0.945i)17-s + (−0.540 − 0.841i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.999 + 0.0109i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (258, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.999 + 0.0109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03359606892 - 6.122521638i\)
\(L(\frac12)\) \(\approx\) \(-0.03359606892 - 6.122521638i\)
\(L(1)\) \(\approx\) \(1.718627325 - 2.003474685i\)
\(L(1)\) \(\approx\) \(1.718627325 - 2.003474685i\)

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 \( 1 + (0.618 - 0.786i)T \)
5 \( 1 + (0.540 - 0.841i)T \)
7 \( 1 + (0.0475 - 0.998i)T \)
11 \( 1 + (0.998 - 0.0475i)T \)
17 \( 1 + (-0.327 + 0.945i)T \)
19 \( 1 + (0.235 + 0.971i)T \)
23 \( 1 + (-0.690 - 0.723i)T \)
29 \( 1 + (0.458 + 0.888i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.928 - 0.371i)T \)
43 \( 1 + (-0.458 + 0.888i)T \)
47 \( 1 + (0.989 - 0.142i)T \)
53 \( 1 + (0.142 - 0.989i)T \)
59 \( 1 + (0.786 - 0.618i)T \)
61 \( 1 + (-0.0950 - 0.995i)T \)
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 \( 1 + (0.654 + 0.755i)T \)
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

−21.83544548216735531808198087384, −20.93273567482093945536919099600, −20.12592996341730531023962402594, −19.405151975227240553614634930102, −18.3738214561939747172890387292, −17.46900309048387178387831493034, −16.6153133269378484749232080732, −15.62359098255112695364417991635, −15.216674126962293421967770204310, −14.55820346905659652714742904049, −13.75392222368007336850741803295, −13.35002939089731139726851516160, −11.80910762535435326156124306420, −11.546376227536149169659096478782, −10.45671301717223952242496633116, −9.45780429839258674144113859875, −8.86975458790889019123223845535, −7.71458787415320945256527344860, −6.84885456791651492649381093409, −5.954824947071997388774784142545, −5.20804386808969940465012197525, −4.27435309017807294389513159079, −3.3179980563442158081249299730, −2.61081960386144242595837663953, −1.909528531630984807030555086156, 0.70833607730875831191058408898, 1.52114887293734148121647262277, 2.10963634685740744103589235884, 3.615005398081694739851811084494, 3.97796326959383187706633752127, 5.15271792187356242498746359511, 6.23213829995201129132697810783, 6.71393160386154453066097524292, 7.79147434894899085984424361019, 8.66901448140898465599869859437, 9.66285463949843028131983752313, 10.44744476777746600489380492565, 11.51685202676505612112293494658, 12.51243158141418887871073997334, 12.759796756032202339982754732094, 13.719030277239332930799203131050, 14.265198156373722926783221175129, 14.734429500210637444108018190259, 16.10282757412925184238720948069, 16.762003139980555389027733913174, 17.5731086665511992252529339098, 18.564986304376057245076676281171, 19.62499339839500992179349481915, 20.04636033344623687834335051377, 20.50158318473936057644672908919

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