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Properties

Label 1-137-137.88-r0-0-0
Degree $1$
Conductor $137$
Sign $0.0188 + 0.999i$
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  + (0.445 + 0.895i)2-s + (0.739 + 0.673i)3-s + (−0.602 + 0.798i)4-s + (0.445 − 0.895i)5-s + (−0.273 + 0.961i)6-s + (0.932 − 0.361i)7-s + (−0.982 − 0.183i)8-s + (0.0922 + 0.995i)9-s + 10-s + (−0.602 + 0.798i)11-s + (−0.982 + 0.183i)12-s + (0.932 − 0.361i)13-s + (0.739 + 0.673i)14-s + (0.932 − 0.361i)15-s + (−0.273 − 0.961i)16-s + (−0.982 + 0.183i)17-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)2-s + (0.739 + 0.673i)3-s + (−0.602 + 0.798i)4-s + (0.445 − 0.895i)5-s + (−0.273 + 0.961i)6-s + (0.932 − 0.361i)7-s + (−0.982 − 0.183i)8-s + (0.0922 + 0.995i)9-s + 10-s + (−0.602 + 0.798i)11-s + (−0.982 + 0.183i)12-s + (0.932 − 0.361i)13-s + (0.739 + 0.673i)14-s + (0.932 − 0.361i)15-s + (−0.273 − 0.961i)16-s + (−0.982 + 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0188 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0188 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.0188 + 0.999i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.0188 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.199791452 + 1.222618738i\)
\(L(\frac12)\) \(\approx\) \(1.199791452 + 1.222618738i\)
\(L(1)\) \(\approx\) \(1.320016592 + 0.8812723646i\)
\(L(1)\) \(\approx\) \(1.320016592 + 0.8812723646i\)

Euler product

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

−28.5600694843356689661502180622, −27.23508344068746624782034444133, −26.32170056850921682255149826375, −25.24881758678673474731627090899, −23.97022657492028993897382231327, −23.46168223501364372284788116301, −21.823741634582309884371773274824, −21.34632262559900264272652586964, −20.25800621963756559029693610366, −19.15054027769621476362546119441, −18.2983510597557287091439403145, −17.83408728158725003586638416600, −15.45510718267196380630343756569, −14.53041475687043181100123460602, −13.715289026403153254558372749193, −12.98926044933783080552958681551, −11.41144612339429088237591956465, −10.85825582280027617565310861385, −9.251676778165278403217087606073, −8.336357764508261454236860684705, −6.71462378752643619042054937973, −5.52591036874432547688789479793, −3.77895705722435178496337038010, −2.58775661437882625105583042180, −1.674078370031674750677535383, 2.18096274002173816946456879207, 4.15381414110244967948193480022, 4.68878303657726643620966926858, 5.99514798043239549726957338205, 7.824203457587305684944178030892, 8.41525265544945395319984780769, 9.516408228605398355886338283739, 10.89007394366990710858696695735, 12.74642039935565798139985543357, 13.45185302565276168361387847177, 14.53381207738029597536931163665, 15.38028668251907299805387173941, 16.36807490938937783467315530350, 17.3223563077533260097265500631, 18.31664668407151190240926646730, 20.26207415827064581797897972206, 20.80408189091696367424096954076, 21.5941707249454141720693063111, 22.89557402449276916920344866305, 23.985282622319895679244584652709, 24.85145446579617207530371606188, 25.65860736413851618226025071975, 26.520820677430035528861812903020, 27.58152273411807631747227287650, 28.35745088892898540268624947789

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