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Properties

Label 1-1157-1157.1099-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.759 - 0.650i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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 + (0.992 + 0.118i)3-s + (0.786 + 0.618i)4-s + (0.841 − 0.540i)5-s + (−0.899 − 0.436i)6-s + (−0.739 − 0.672i)7-s + (−0.540 − 0.841i)8-s + (0.971 + 0.235i)9-s + (−0.971 + 0.235i)10-s + (0.998 + 0.0475i)11-s + (0.707 + 0.707i)12-s + (0.479 + 0.877i)14-s + (0.899 − 0.436i)15-s + (0.235 + 0.971i)16-s + (0.945 − 0.327i)17-s + (−0.841 − 0.540i)18-s + ⋯
L(s)  = 1  + (−0.945 − 0.327i)2-s + (0.992 + 0.118i)3-s + (0.786 + 0.618i)4-s + (0.841 − 0.540i)5-s + (−0.899 − 0.436i)6-s + (−0.739 − 0.672i)7-s + (−0.540 − 0.841i)8-s + (0.971 + 0.235i)9-s + (−0.971 + 0.235i)10-s + (0.998 + 0.0475i)11-s + (0.707 + 0.707i)12-s + (0.479 + 0.877i)14-s + (0.899 − 0.436i)15-s + (0.235 + 0.971i)16-s + (0.945 − 0.327i)17-s + (−0.841 − 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.759 - 0.650i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.759 - 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.714461070 - 0.6342410101i\)
\(L(\frac12)\) \(\approx\) \(1.714461070 - 0.6342410101i\)
\(L(1)\) \(\approx\) \(1.189141563 - 0.2730869511i\)
\(L(1)\) \(\approx\) \(1.189141563 - 0.2730869511i\)

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 \( 1 + (0.992 + 0.118i)T \)
5 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (-0.739 - 0.672i)T \)
11 \( 1 + (0.998 + 0.0475i)T \)
17 \( 1 + (0.945 - 0.327i)T \)
19 \( 1 + (0.853 - 0.520i)T \)
23 \( 1 + (0.0237 + 0.999i)T \)
29 \( 1 + (-0.952 + 0.304i)T \)
31 \( 1 + (0.479 + 0.877i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (-0.393 + 0.919i)T \)
43 \( 1 + (0.952 + 0.304i)T \)
47 \( 1 + (0.142 - 0.989i)T \)
53 \( 1 + (-0.989 + 0.142i)T \)
59 \( 1 + (0.118 + 0.992i)T \)
61 \( 1 + (-0.636 - 0.771i)T \)
67 \( 1 + (-0.618 - 0.786i)T \)
71 \( 1 + (-0.888 + 0.458i)T \)
73 \( 1 + (0.281 - 0.959i)T \)
79 \( 1 + (0.281 - 0.959i)T \)
83 \( 1 + (-0.0713 - 0.997i)T \)
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

−20.99379503728754729593631914931, −20.60083196491885522406062558536, −19.5115367221286964615896216423, −18.89508435751608577025461004228, −18.58935151774261787755535166413, −17.60497596920130624126072871176, −16.772509921272463027909558420049, −15.98569509008738135656331291962, −15.10164075253466628120320789068, −14.44428076654863549034332602385, −13.946607095956655046358283824453, −12.736631227508943723763472678477, −11.97897869655395163883522340686, −10.76990510682580942232045616474, −9.84670059816366844247803387970, −9.45977549562460258410198733033, −8.81081475226130376789273023446, −7.80491915150407285138401072953, −7.00332643047457493111186153753, −6.200322699415830778479515209029, −5.57304074894382289572852910729, −3.80081853021862656455100882681, −2.86987843348222209166848630359, −2.12593211833998642978237593016, −1.19748319547738669948184203387, 1.11550587538461515104156962308, 1.63739257172637188654479474309, 3.01205556804260091535342558298, 3.45932476926411290704610346944, 4.68229185469924604154176896333, 6.067859139162382740781526822830, 7.02090934913363731022388128267, 7.64322245072357008660152285774, 8.72678398843314761926533820096, 9.44160250287816986466269288841, 9.72766961755486909326204111106, 10.53984296849444206247455046867, 11.76453997008681205818065756880, 12.588847912787735302038326619047, 13.46157497569197516679669105164, 13.97926895732483327582235076867, 15.06181582035430297902434858762, 16.07096027266656812987136218319, 16.60048975049916987263005654921, 17.33227006490425270074086309853, 18.17592608194895907975426672372, 19.078017616425854890937383972854, 19.73399669695014339721777778564, 20.2677332382502457264465866380, 20.87917678412333018331195890639

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