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Properties

Label 1-1157-1157.106-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.949 + 0.313i$
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  + (0.458 + 0.888i)2-s + (−0.814 + 0.580i)3-s + (−0.580 + 0.814i)4-s + (−0.989 − 0.142i)5-s + (−0.888 − 0.458i)6-s + (−0.928 + 0.371i)7-s + (−0.989 − 0.142i)8-s + (0.327 − 0.945i)9-s + (−0.327 − 0.945i)10-s + (−0.371 + 0.928i)11-s − i·12-s + (−0.755 − 0.654i)14-s + (0.888 − 0.458i)15-s + (−0.327 − 0.945i)16-s + (−0.888 − 0.458i)17-s + (0.989 − 0.142i)18-s + ⋯
L(s)  = 1  + (0.458 + 0.888i)2-s + (−0.814 + 0.580i)3-s + (−0.580 + 0.814i)4-s + (−0.989 − 0.142i)5-s + (−0.888 − 0.458i)6-s + (−0.928 + 0.371i)7-s + (−0.989 − 0.142i)8-s + (0.327 − 0.945i)9-s + (−0.327 − 0.945i)10-s + (−0.371 + 0.928i)11-s − i·12-s + (−0.755 − 0.654i)14-s + (0.888 − 0.458i)15-s + (−0.327 − 0.945i)16-s + (−0.888 − 0.458i)17-s + (0.989 − 0.142i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.949 + 0.313i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.949 + 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07133086810 + 0.4433936050i\)
\(L(\frac12)\) \(\approx\) \(0.07133086810 + 0.4433936050i\)
\(L(1)\) \(\approx\) \(0.4558033185 + 0.3694159189i\)
\(L(1)\) \(\approx\) \(0.4558033185 + 0.3694159189i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.458 + 0.888i)T \)
3 \( 1 + (-0.814 + 0.580i)T \)
5 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-0.928 + 0.371i)T \)
11 \( 1 + (-0.371 + 0.928i)T \)
17 \( 1 + (-0.888 - 0.458i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
23 \( 1 + (-0.189 - 0.981i)T \)
29 \( 1 + (-0.618 + 0.786i)T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.995 - 0.0950i)T \)
43 \( 1 + (0.618 + 0.786i)T \)
47 \( 1 + (0.909 - 0.415i)T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (0.580 - 0.814i)T \)
61 \( 1 + (-0.690 + 0.723i)T \)
67 \( 1 + (0.814 - 0.580i)T \)
71 \( 1 + (-0.618 - 0.786i)T \)
73 \( 1 + (-0.755 - 0.654i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.371 - 0.928i)T \)
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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

−20.682180746571597032708480608566, −19.77591954532732031515692404082, −19.12923782368691880688217258818, −18.86340278386387006764128371875, −17.85553204963054635951015915436, −16.93828002320724380399589686841, −15.93047037706622691315374772039, −15.52202268361257513409328671392, −14.14258672481972048541229967221, −13.46584962781430660056969011849, −12.776346340955554873136678293012, −12.06756383512408673278115202938, −11.38091680957104632962433075170, −10.70360666452295042468354949724, −10.05410279539318620741951138177, −8.81425337179050312769494279135, −7.852809676498675000617901866815, −6.86690074485162086112667256803, −6.02549420152642621139796214791, −5.283271760120145843381770809795, −4.062841900509315474347047866092, −3.52453271686127934978974938282, −2.41562569661280050985239485030, −1.14953403639352386972672456363, −0.223011850606421602489944890081, 0.43549640273831713400103945222, 2.70677651934480550463691549409, 3.61720190944476342132006561541, 4.61189431614131537245585318553, 4.93002441181540909089264460758, 6.14706584652823928775751735333, 6.83670797127685598472990601751, 7.49382254260103413034521272370, 8.758613101857085933451370682689, 9.326673634108615203078062523588, 10.351788811010910955054673743638, 11.43453198485518813303055881830, 12.17313791876418535306505095205, 12.71767772092199553404719589889, 13.55486270379533928800749173880, 14.99130740510095797014697478584, 15.3205516603766874225858222671, 15.94414021597654024633495108136, 16.51407435207031076683204153642, 17.34998889643731554956752315581, 18.18176898952907956457693728798, 18.86356562347809726957251102172, 20.17502755424312388377010075216, 20.65158369697285392073874197989, 21.934618728800956636263278203827

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