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Properties

Label 1-1157-1157.579-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.419 + 0.907i$
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  + (0.998 + 0.0475i)2-s + (−0.995 + 0.0950i)3-s + (0.995 + 0.0950i)4-s + (−0.989 − 0.142i)5-s + (−0.998 + 0.0475i)6-s + (−0.618 + 0.786i)7-s + (0.989 + 0.142i)8-s + (0.981 − 0.189i)9-s + (−0.981 − 0.189i)10-s + (0.618 + 0.786i)11-s − 12-s + (−0.654 + 0.755i)14-s + (0.998 + 0.0475i)15-s + (0.981 + 0.189i)16-s + (−0.0475 − 0.998i)17-s + (0.989 − 0.142i)18-s + ⋯
L(s)  = 1  + (0.998 + 0.0475i)2-s + (−0.995 + 0.0950i)3-s + (0.995 + 0.0950i)4-s + (−0.989 − 0.142i)5-s + (−0.998 + 0.0475i)6-s + (−0.618 + 0.786i)7-s + (0.989 + 0.142i)8-s + (0.981 − 0.189i)9-s + (−0.981 − 0.189i)10-s + (0.618 + 0.786i)11-s − 12-s + (−0.654 + 0.755i)14-s + (0.998 + 0.0475i)15-s + (0.981 + 0.189i)16-s + (−0.0475 − 0.998i)17-s + (0.989 − 0.142i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.419 + 0.907i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.419 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.983014307 + 1.267667077i\)
\(L(\frac12)\) \(\approx\) \(1.983014307 + 1.267667077i\)
\(L(1)\) \(\approx\) \(1.271524747 + 0.2544155977i\)
\(L(1)\) \(\approx\) \(1.271524747 + 0.2544155977i\)

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 \( 1 + (-0.995 + 0.0950i)T \)
5 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-0.618 + 0.786i)T \)
11 \( 1 + (0.618 + 0.786i)T \)
17 \( 1 + (-0.0475 - 0.998i)T \)
19 \( 1 + (0.189 + 0.981i)T \)
23 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (0.928 - 0.371i)T \)
31 \( 1 + (-0.755 - 0.654i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.814 + 0.580i)T \)
43 \( 1 + (-0.928 - 0.371i)T \)
47 \( 1 + (0.909 - 0.415i)T \)
53 \( 1 + (0.415 - 0.909i)T \)
59 \( 1 + (-0.0950 + 0.995i)T \)
61 \( 1 + (0.235 - 0.971i)T \)
67 \( 1 + (-0.0950 - 0.995i)T \)
71 \( 1 + (-0.371 + 0.928i)T \)
73 \( 1 + (0.755 + 0.654i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (0.540 + 0.841i)T \)
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

−21.28724252313889392275930778295, −20.03908842944165453198808260935, −19.49883858890581613885791391684, −19.003782279966369838246544796105, −17.59233933048918881351418897251, −16.900378889151588966722774149171, −16.071174148982201377738724761776, −15.745371851518856035054982367703, −14.73191468386336401255722162271, −13.77113451679238461413242810977, −13.03388939355986994755602237313, −12.28714223694526972473003873865, −11.61106490382809545817447185699, −10.81538255295673125678795296096, −10.47635834730178580896353843307, −9.01770593689916195165886334896, −7.658084746446351513606931172422, −7.000997132596228789024841374641, −6.40221140650002454669494055029, −5.47712516297166814826580353072, −4.48248221246492070075650076416, −3.79112808628253835782326130968, −3.10915639228128155166751485838, −1.42869015914995058725877159510, −0.52984481166477214099070368864, 0.80459389125781711815223301148, 2.138129785685441981646503583915, 3.31898592728734721237647632794, 4.17001977778363241059250785528, 4.89209011286824273311893483825, 5.69289060124263795921972189103, 6.677514932626135985140010364699, 7.11601539097536888441678165010, 8.250463342298757913193615468563, 9.51440238505799453495994471263, 10.38159753488158885593587346738, 11.397186184192468914248264255385, 12.01775206686974660434372741627, 12.3422142982735916297506184106, 13.120602665249559673303341334418, 14.36388168360745172012843499593, 15.18594153585681141994602262631, 15.73044304640145084284683074927, 16.4283670937659990401805481154, 16.97046206413461337441631317287, 18.27384796331760680060670177021, 18.907050215945757520862869897640, 19.86829962561102472210406734201, 20.57269504414907145563923884055, 21.4121170967800630375232184828

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