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Properties

Label 1-137-137.39-r0-0-0
Degree $1$
Conductor $137$
Sign $0.936 + 0.349i$
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  + (0.850 − 0.526i)2-s + (0.361 + 0.932i)3-s + (0.445 − 0.895i)4-s + (−0.526 + 0.850i)5-s + (0.798 + 0.602i)6-s + (0.982 + 0.183i)7-s + (−0.0922 − 0.995i)8-s + (−0.739 + 0.673i)9-s + i·10-s + (−0.445 + 0.895i)11-s + (0.995 + 0.0922i)12-s + (0.183 − 0.982i)13-s + (0.932 − 0.361i)14-s + (−0.982 − 0.183i)15-s + (−0.602 − 0.798i)16-s + (−0.0922 + 0.995i)17-s + ⋯
L(s)  = 1  + (0.850 − 0.526i)2-s + (0.361 + 0.932i)3-s + (0.445 − 0.895i)4-s + (−0.526 + 0.850i)5-s + (0.798 + 0.602i)6-s + (0.982 + 0.183i)7-s + (−0.0922 − 0.995i)8-s + (−0.739 + 0.673i)9-s + i·10-s + (−0.445 + 0.895i)11-s + (0.995 + 0.0922i)12-s + (0.183 − 0.982i)13-s + (0.932 − 0.361i)14-s + (−0.982 − 0.183i)15-s + (−0.602 − 0.798i)16-s + (−0.0922 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.936 + 0.349i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.936 + 0.349i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.800010581 + 0.3250992782i\)
\(L(\frac12)\) \(\approx\) \(1.800010581 + 0.3250992782i\)
\(L(1)\) \(\approx\) \(1.689683573 + 0.1358583259i\)
\(L(1)\) \(\approx\) \(1.689683573 + 0.1358583259i\)

Euler product

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

−28.782582021121265625497354742770, −27.17518802546568371957169191619, −26.31145633032120139239962680284, −24.864963035723553226667732579, −24.47528712192736359005979242635, −23.660454300559100406548861610631, −22.944486368293073206236780442725, −21.08815280443335833407750680748, −20.79833807720939027984529742141, −19.4709151887990102031940596273, −18.28368585115670363491419478462, −17.0494654519709395438523014565, −16.19630289057725810816105287636, −14.91925829795753969423672664895, −13.83985256475603267132413406098, −13.26602333085430536098524094183, −11.85532628690464600102092483820, −11.47562956811773842360873340246, −8.86513755502317814420995137471, −8.04700853148512845324059836636, −7.21357779256564923530126407814, −5.75400325411960199735006424769, −4.626008574318302216237727803755, −3.24678202085497465867338991843, −1.56045631379400611202648248207, 2.241408676689771159536160145643, 3.33158120132840268170924567328, 4.5020845395986415060444913121, 5.43844638848160288290545565015, 7.16420078640145701010080815229, 8.522030893022391071866407502937, 10.21738146664264610188802077387, 10.7691547950049114949500257779, 11.74802225321990452500483413816, 13.15180930740078614769323383469, 14.466137501459584721202263150259, 15.13879646585877890841199920112, 15.59680052356864497379086140703, 17.46413204649497910737214584106, 18.72333496319244132029598066888, 19.91630569274778461030653008846, 20.61864397400323487557725251370, 21.57875325893330151035339584699, 22.460786752660675194692593105, 23.22060547227205010026040792814, 24.408541405758074873311691546682, 25.64493731806644010212991122974, 26.66499820062573738295270562741, 27.84807096026244032325187468362, 28.21723637456654670440328207225

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