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Properties

Label 1-1157-1157.15-r0-0-0
Degree $1$
Conductor $1157$
Sign $-0.378 + 0.925i$
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  + (0.998 − 0.0475i)2-s + (0.636 + 0.771i)3-s + (0.995 − 0.0950i)4-s + (0.142 + 0.989i)5-s + (0.672 + 0.739i)6-s + (−0.118 + 0.992i)7-s + (0.989 − 0.142i)8-s + (−0.189 + 0.981i)9-s + (0.189 + 0.981i)10-s + (−0.618 + 0.786i)11-s + (0.707 + 0.707i)12-s + (−0.0713 + 0.997i)14-s + (−0.672 + 0.739i)15-s + (0.981 − 0.189i)16-s + (0.998 + 0.0475i)17-s + (−0.142 + 0.989i)18-s + ⋯
L(s)  = 1  + (0.998 − 0.0475i)2-s + (0.636 + 0.771i)3-s + (0.995 − 0.0950i)4-s + (0.142 + 0.989i)5-s + (0.672 + 0.739i)6-s + (−0.118 + 0.992i)7-s + (0.989 − 0.142i)8-s + (−0.189 + 0.981i)9-s + (0.189 + 0.981i)10-s + (−0.618 + 0.786i)11-s + (0.707 + 0.707i)12-s + (−0.0713 + 0.997i)14-s + (−0.672 + 0.739i)15-s + (0.981 − 0.189i)16-s + (0.998 + 0.0475i)17-s + (−0.142 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.378 + 0.925i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.087055343 + 3.109621196i\)
\(L(\frac12)\) \(\approx\) \(2.087055343 + 3.109621196i\)
\(L(1)\) \(\approx\) \(2.025939487 + 1.252669885i\)
\(L(1)\) \(\approx\) \(2.025939487 + 1.252669885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.998 - 0.0475i)T \)
3 \( 1 + (0.636 + 0.771i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (-0.118 + 0.992i)T \)
11 \( 1 + (-0.618 + 0.786i)T \)
17 \( 1 + (0.998 + 0.0475i)T \)
19 \( 1 + (-0.560 - 0.828i)T \)
23 \( 1 + (0.436 - 0.899i)T \)
29 \( 1 + (0.919 - 0.393i)T \)
31 \( 1 + (0.0713 - 0.997i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (0.986 + 0.165i)T \)
43 \( 1 + (-0.919 - 0.393i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (0.909 - 0.415i)T \)
59 \( 1 + (-0.771 - 0.636i)T \)
61 \( 1 + (-0.520 + 0.853i)T \)
67 \( 1 + (-0.0950 + 0.995i)T \)
71 \( 1 + (-0.928 + 0.371i)T \)
73 \( 1 + (-0.755 + 0.654i)T \)
79 \( 1 + (0.755 - 0.654i)T \)
83 \( 1 + (0.977 - 0.212i)T \)
97 \( 1 + (0.618 + 0.786i)T \)
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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

−21.14894123312748672446554190131, −20.39654364948505773063524458182, −19.58582135862157040878042914842, −19.1675961304047313715893174267, −17.87630641551382398839136982102, −16.95524685423062886322438757920, −16.36133704045751659457895619757, −15.56070575230582399512152331701, −14.44884075597738408268504933939, −13.82418323129466288693224431016, −13.39319807289171972028764294607, −12.51060654198610299910913235348, −12.1237549270528543814543768739, −10.89014048237463324255003549153, −10.072692399400973799041830239106, −8.87261374083293763989854860855, −7.945788538111800915741403338793, −7.477200583860373637825509562246, −6.38587232363811293205303653769, −5.621817653686181811635941146539, −4.67536161601843278539297409162, −3.61719090549975672103736733825, −3.04082921936868231173605307888, −1.68506029673187799294135990094, −1.001425054223907734406795407218, 2.0941069069979512365592053473, 2.59216766412444984494259803142, 3.269124913131631766080859677689, 4.33781248693712638804527326434, 5.14230251407005591276352698089, 5.9665963403528586665196342819, 6.94134637214290598990072501686, 7.79782184503208830613195649286, 8.789812394279259810834426493003, 10.015443715405079096024848005823, 10.347566199507009087609827446141, 11.33260222322865430387217257812, 12.15343751166483659762564786461, 13.089294724559772478542930416566, 13.8415440361113484721243056522, 14.83143778176668154765926521712, 15.01524186027321157176970693064, 15.675835240207879386283203507132, 16.54242309151036581400497690994, 17.64338684635574586435451103694, 18.81700605511656649776837329466, 19.210593172043217371819614340642, 20.244084817900068007871027094799, 21.03632793546506157817186263964, 21.50829800918222138430203551644

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