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Properties

Label 1-137-137.25-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: $\chi_{137} (25, \cdot )$
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

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

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