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Properties

Label 1-137-137.11-r0-0-0
Degree $1$
Conductor $137$
Sign $0.796 - 0.605i$
Analytic cond. $0.636225$
Root an. cond. $0.636225$
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  + (0.982 − 0.183i)2-s + (0.798 − 0.602i)3-s + (0.932 − 0.361i)4-s + (−0.183 + 0.982i)5-s + (0.673 − 0.739i)6-s + (−0.445 − 0.895i)7-s + (0.850 − 0.526i)8-s + (0.273 − 0.961i)9-s + i·10-s + (−0.932 + 0.361i)11-s + (0.526 − 0.850i)12-s + (−0.895 + 0.445i)13-s + (−0.602 − 0.798i)14-s + (0.445 + 0.895i)15-s + (0.739 − 0.673i)16-s + (0.850 + 0.526i)17-s + ⋯
L(s)  = 1  + (0.982 − 0.183i)2-s + (0.798 − 0.602i)3-s + (0.932 − 0.361i)4-s + (−0.183 + 0.982i)5-s + (0.673 − 0.739i)6-s + (−0.445 − 0.895i)7-s + (0.850 − 0.526i)8-s + (0.273 − 0.961i)9-s + i·10-s + (−0.932 + 0.361i)11-s + (0.526 − 0.850i)12-s + (−0.895 + 0.445i)13-s + (−0.602 − 0.798i)14-s + (0.445 + 0.895i)15-s + (0.739 − 0.673i)16-s + (0.850 + 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.796 - 0.605i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.796 - 0.605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.071981741 - 0.6979332539i\)
\(L(\frac12)\) \(\approx\) \(2.071981741 - 0.6979332539i\)
\(L(1)\) \(\approx\) \(1.938123596 - 0.4600708015i\)
\(L(1)\) \(\approx\) \(1.938123596 - 0.4600708015i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (0.982 - 0.183i)T \)
3 \( 1 + (0.798 - 0.602i)T \)
5 \( 1 + (-0.183 + 0.982i)T \)
7 \( 1 + (-0.445 - 0.895i)T \)
11 \( 1 + (-0.932 + 0.361i)T \)
13 \( 1 + (-0.895 + 0.445i)T \)
17 \( 1 + (0.850 + 0.526i)T \)
19 \( 1 + (-0.0922 + 0.995i)T \)
23 \( 1 + (0.673 + 0.739i)T \)
29 \( 1 + (-0.673 - 0.739i)T \)
31 \( 1 + (-0.995 + 0.0922i)T \)
37 \( 1 - T \)
41 \( 1 - iT \)
43 \( 1 + (0.995 + 0.0922i)T \)
47 \( 1 + (0.961 + 0.273i)T \)
53 \( 1 + (-0.995 - 0.0922i)T \)
59 \( 1 + (-0.273 + 0.961i)T \)
61 \( 1 + (0.273 + 0.961i)T \)
67 \( 1 + (0.895 - 0.445i)T \)
71 \( 1 + (-0.361 + 0.932i)T \)
73 \( 1 + (0.445 - 0.895i)T \)
79 \( 1 + (-0.798 - 0.602i)T \)
83 \( 1 + (-0.526 - 0.850i)T \)
89 \( 1 + (0.183 - 0.982i)T \)
97 \( 1 + (0.361 + 0.932i)T \)
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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

−28.651182226751401304524851619360, −27.65976841962654813177353282574, −26.352832057466981702655257581170, −25.35506236302514824374457817884, −24.69288065191875783916047405799, −23.764544940837951756179267564072, −22.394452511532344347852880608031, −21.57601936317813420269623896108, −20.73527451141350575977061261449, −19.94943695446875469356973944903, −18.85756364187766887172505745758, −16.90155120432124089935330157735, −15.97275569429190009312851620290, −15.37000248714374164571833578024, −14.306082586628134718291580752892, −13.013569501286454308541031047180, −12.5032681232611110574762974144, −11.00210114104570870151101846988, −9.555098043319206517615770329532, −8.453458137893226119175301315489, −7.37696955286965869655562964341, −5.41008499575876349893528744419, −4.904222240161818730156943042470, −3.32900667865979175528573510888, −2.3895991826787546964786722347, 1.92063797141014760484742385133, 3.12218093991697948630942403005, 3.994516722410478190541534835280, 5.854312086441384067468397087246, 7.287458943675306818583675995534, 7.49959611532142168771808468512, 9.80783807648440688894157847199, 10.66260532059741564553612316132, 12.122943322239688933845540743604, 13.02305706762173621850831030994, 14.062846999214036784489356750390, 14.69405734116010725205512914820, 15.70554583535751168033078773401, 17.191592476230476331480548413511, 18.85598970143517089444645868371, 19.30034973713781154001924210553, 20.455983104865591902252876209064, 21.29090795210259636372293561766, 22.640685777619733452157026508378, 23.39433694713962745226452965894, 24.133793098801661194694435894709, 25.54637371236213754154203438288, 26.02248929789274367992568106758, 27.15735737654488019804795270963, 29.06864859993452510478758119830

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