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Properties

Label 1-1117-1117.29-r1-0-0
Degree $1$
Conductor $1117$
Sign $-0.992 - 0.124i$
Analytic cond. $120.038$
Root an. cond. $120.038$
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  + (0.145 − 0.989i)2-s + (−0.184 − 0.982i)3-s + (−0.957 − 0.288i)4-s + (−0.266 + 0.963i)5-s + (−0.999 + 0.0394i)6-s + (−0.927 − 0.373i)7-s + (−0.425 + 0.905i)8-s + (−0.931 + 0.363i)9-s + (0.914 + 0.404i)10-s + (−0.866 − 0.5i)11-s + (−0.106 + 0.994i)12-s + (−0.0506 + 0.998i)13-s + (−0.504 + 0.863i)14-s + (0.996 + 0.0843i)15-s + (0.833 + 0.552i)16-s + (0.212 + 0.977i)17-s + ⋯
L(s)  = 1  + (0.145 − 0.989i)2-s + (−0.184 − 0.982i)3-s + (−0.957 − 0.288i)4-s + (−0.266 + 0.963i)5-s + (−0.999 + 0.0394i)6-s + (−0.927 − 0.373i)7-s + (−0.425 + 0.905i)8-s + (−0.931 + 0.363i)9-s + (0.914 + 0.404i)10-s + (−0.866 − 0.5i)11-s + (−0.106 + 0.994i)12-s + (−0.0506 + 0.998i)13-s + (−0.504 + 0.863i)14-s + (0.996 + 0.0843i)15-s + (0.833 + 0.552i)16-s + (0.212 + 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1117\)
Sign: $-0.992 - 0.124i$
Analytic conductor: \(120.038\)
Root analytic conductor: \(120.038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1117} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1117,\ (1:\ ),\ -0.992 - 0.124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008440114293 - 0.1348594359i\)
\(L(\frac12)\) \(\approx\) \(0.008440114293 - 0.1348594359i\)
\(L(1)\) \(\approx\) \(0.5245220724 - 0.2817056088i\)
\(L(1)\) \(\approx\) \(0.5245220724 - 0.2817056088i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1117 \( 1 \)
good2 \( 1 + (0.145 - 0.989i)T \)
3 \( 1 + (-0.184 - 0.982i)T \)
5 \( 1 + (-0.266 + 0.963i)T \)
7 \( 1 + (-0.927 - 0.373i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.0506 + 0.998i)T \)
17 \( 1 + (0.212 + 0.977i)T \)
19 \( 1 + (-0.747 + 0.664i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.616 + 0.787i)T \)
31 \( 1 + (-0.996 - 0.0787i)T \)
37 \( 1 + (-0.151 + 0.988i)T \)
41 \( 1 + (-0.827 + 0.561i)T \)
43 \( 1 + (0.980 - 0.195i)T \)
47 \( 1 + (-0.963 - 0.266i)T \)
53 \( 1 + (0.0955 - 0.995i)T \)
59 \( 1 + (0.524 + 0.851i)T \)
61 \( 1 + (-0.480 - 0.877i)T \)
67 \( 1 + (0.634 + 0.773i)T \)
71 \( 1 + (-0.256 + 0.966i)T \)
73 \( 1 + (0.552 - 0.833i)T \)
79 \( 1 + (-0.00563 - 0.999i)T \)
83 \( 1 + (-0.807 + 0.589i)T \)
89 \( 1 + (-0.612 + 0.790i)T \)
97 \( 1 + (-0.162 + 0.986i)T \)
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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

−21.57805948238411095981098715886, −20.94943111092817940387648580081, −20.13698509465658906431006794231, −19.246932811138401654071809581860, −18.17671842066870287519212810982, −17.35125029049578088967117754405, −16.706438698383116176517098911256, −15.9584598615338728631851873566, −15.47280243070240578156760276394, −15.025847531874039117608678491059, −13.74305240350378686518110853329, −12.83674768934424253638962437864, −12.51747441246583803434888722464, −11.23187109367258881077580979503, −10.11631660412927123489843500372, −9.36733010798176362456519300172, −8.89536564666501327590118996258, −7.9052944124264901240696758328, −7.04190829528319642273771646869, −5.732958505832999984317745178597, −5.33744305918397031231757631049, −4.57456811707877213149647834693, −3.62140680787645009762957458075, −2.71320710606442135982346010709, −0.46063936903599043821567677824, 0.05786538438872997135179708755, 1.43067256973054394027353031475, 2.37324959424650602582025765317, 3.21381017264859384719741364406, 3.91110453688119263282888389624, 5.31921544323772348484660821245, 6.26980387810486713691114172248, 6.91492101963606246380020876884, 7.97252955043272258419691489866, 8.78350198383573298933770624274, 9.95269220690327427371031330215, 10.74035538274274423454707647780, 11.20617448638613532520935439694, 12.2048925011093140271393485278, 12.9365301458951742625256081763, 13.433798214153744215819627752733, 14.37130737235287226631523052507, 14.94855392005985516176723335752, 16.4107026769279545013616377435, 17.057855478008979508603731910963, 18.1394076265121657477629767096, 18.91500494806719339977774665404, 18.98130493394414270688554044023, 19.76573918488503604897553914022, 20.74986345627489390602232367447

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