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Properties

Label 1-1157-1157.1050-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.243 + 0.969i$
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  + (0.786 + 0.618i)2-s + (0.971 + 0.235i)3-s + (0.235 + 0.971i)4-s + (0.415 − 0.909i)5-s + (0.618 + 0.786i)6-s + (0.0950 + 0.995i)7-s + (−0.415 + 0.909i)8-s + (0.888 + 0.458i)9-s + (0.888 − 0.458i)10-s + (0.995 + 0.0950i)11-s + i·12-s + (−0.540 + 0.841i)14-s + (0.618 − 0.786i)15-s + (−0.888 + 0.458i)16-s + (0.786 − 0.618i)17-s + (0.415 + 0.909i)18-s + ⋯
L(s)  = 1  + (0.786 + 0.618i)2-s + (0.971 + 0.235i)3-s + (0.235 + 0.971i)4-s + (0.415 − 0.909i)5-s + (0.618 + 0.786i)6-s + (0.0950 + 0.995i)7-s + (−0.415 + 0.909i)8-s + (0.888 + 0.458i)9-s + (0.888 − 0.458i)10-s + (0.995 + 0.0950i)11-s + i·12-s + (−0.540 + 0.841i)14-s + (0.618 − 0.786i)15-s + (−0.888 + 0.458i)16-s + (0.786 − 0.618i)17-s + (0.415 + 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.243 + 0.969i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1050, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.243 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.169291498 + 2.472657519i\)
\(L(\frac12)\) \(\approx\) \(3.169291498 + 2.472657519i\)
\(L(1)\) \(\approx\) \(2.228515364 + 1.097750624i\)
\(L(1)\) \(\approx\) \(2.228515364 + 1.097750624i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.786 + 0.618i)T \)
3 \( 1 + (0.971 + 0.235i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (0.0950 + 0.995i)T \)
11 \( 1 + (0.995 + 0.0950i)T \)
17 \( 1 + (0.786 - 0.618i)T \)
19 \( 1 + (0.458 - 0.888i)T \)
23 \( 1 + (-0.998 + 0.0475i)T \)
29 \( 1 + (0.814 - 0.580i)T \)
31 \( 1 + (-0.540 + 0.841i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.690 - 0.723i)T \)
43 \( 1 + (0.814 + 0.580i)T \)
47 \( 1 + (-0.959 - 0.281i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (-0.971 + 0.235i)T \)
61 \( 1 + (-0.189 + 0.981i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (0.580 - 0.814i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (-0.989 + 0.142i)T \)
97 \( 1 + (0.995 - 0.0950i)T \)
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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

−21.159649799345908098988205477412, −20.213008121013285960621031635484, −19.85442406251487706921294462774, −18.933363588498574993338749594595, −18.45203920726756234198811621005, −17.387555571520967056224492460677, −16.332912248817532518085101190708, −15.248838557030595302083634902358, −14.42326938275472419150497750169, −14.16443755914548177055575409054, −13.58580203318737165644851003644, −12.58761145459212696362743527493, −11.81779440455486295627309814115, −10.78623583958840497239908223820, −10.02134190052051541384305810321, −9.62485463936219120856777969142, −8.27200496909540140105475338986, −7.28776932130401132595970477085, −6.58425485104309802852650985223, −5.78720424270826488695908472464, −4.34471526773460294323163382957, −3.61583820015502438520346758256, −3.13106817701085978659111395312, −1.84987054220710359661742299516, −1.31494753143486905778646108726, 1.55456373912601365424031184462, 2.48548021403149342817624388179, 3.39729020675999898700709326346, 4.380723911353412949777404290246, 5.10717758645281001581171546948, 5.89572608576794086670813255954, 6.96249197981906224795936865450, 7.886775254277142451379813821651, 8.81064624498668958670923935949, 9.08440024006256250253574199642, 10.0933156229526737141492977823, 11.72541579327062688158761407261, 12.16777731431792104755304308874, 13.00503530849816763459295648002, 13.90825423955102395278576351351, 14.27902810057176832314886514228, 15.21016128780379447857211872868, 15.95486725760036551204991499155, 16.4026949348988325966670487001, 17.52041422866345563102135669367, 18.1620763364727188316401096555, 19.43824689934953120649646327058, 20.04810973075057532864150597859, 20.90782296142458346058014217750, 21.447991593488702736945428273616

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