Properties

Label 1-137-137.77-r0-0-0
Degree $1$
Conductor $137$
Sign $-0.300 + 0.953i$
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  + (0.932 + 0.361i)2-s + (0.273 + 0.961i)3-s + (0.739 + 0.673i)4-s + (−0.932 + 0.361i)5-s + (−0.0922 + 0.995i)6-s + (−0.602 − 0.798i)7-s + (0.445 + 0.895i)8-s + (−0.850 + 0.526i)9-s − 10-s + (0.739 + 0.673i)11-s + (−0.445 + 0.895i)12-s + (0.602 + 0.798i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 + 0.995i)16-s + (0.445 − 0.895i)17-s + ⋯
L(s)  = 1  + (0.932 + 0.361i)2-s + (0.273 + 0.961i)3-s + (0.739 + 0.673i)4-s + (−0.932 + 0.361i)5-s + (−0.0922 + 0.995i)6-s + (−0.602 − 0.798i)7-s + (0.445 + 0.895i)8-s + (−0.850 + 0.526i)9-s − 10-s + (0.739 + 0.673i)11-s + (−0.445 + 0.895i)12-s + (0.602 + 0.798i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 + 0.995i)16-s + (0.445 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.300 + 0.953i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ -0.300 + 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9852567009 + 1.343438318i\)
\(L(\frac12)\) \(\approx\) \(0.9852567009 + 1.343438318i\)
\(L(1)\) \(\approx\) \(1.286438721 + 0.9090389139i\)
\(L(1)\) \(\approx\) \(1.286438721 + 0.9090389139i\)

Euler product

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

−28.34270669938603727414253719028, −27.617188108427542220097541605733, −25.721576778605101834841081628046, −25.02573662557398448148177840252, −24.008403055524215177933506140699, −23.35221069003219217554017665586, −22.37798824425870859551983429366, −21.226239458432560002596250438881, −19.89765697241221302740054839716, −19.40549637867435313749135815487, −18.59325363472508348861273559232, −16.85929989369427689025439364989, −15.5759326443898454932489604908, −14.8220852990783317156815524199, −13.487417593357753383427990895512, −12.62350677753681202245700217652, −11.968061304516183830156403905550, −10.90544031105277577020784512169, −9.019365277483483821739715567682, −7.93520379964071210880613742715, −6.49325156877105736949739349251, −5.67043986527862519814773272130, −3.83157182308927080108552967221, −2.92411959608529928061459549732, −1.23853079491852993314257825470, 2.763334463025640213508087617715, 4.075986496295229051900514600454, 4.35124679972519331167086297764, 6.28842054219661361049888190832, 7.29519570789059638019308677353, 8.59276937298582033851472697733, 10.119368121934552899461612905059, 11.23142677400897992976606255326, 12.17715005241528054145035872168, 13.69064235588233217948633741921, 14.52632599260166241578976844344, 15.430523990353067522170454889760, 16.342928071655254379114444067041, 17.00475646580919632766999046678, 19.07395873140829405066083956713, 20.11023477513851369325370799391, 20.785641933427883707511869463010, 22.02642290779768538879491573448, 22.995681096913386346859485014094, 23.27678451871411343690582308498, 24.873147040477363822663574196143, 25.93294132525105051286986144997, 26.56275382870275303774178177484, 27.59140178126999107363020154785, 28.81282496732702042590115967259

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