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Properties

Label 1-137-137.76-r0-0-0
Degree $1$
Conductor $137$
Sign $0.358 + 0.933i$
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  + (0.602 − 0.798i)2-s + (−0.995 + 0.0922i)3-s + (−0.273 − 0.961i)4-s + (−0.798 + 0.602i)5-s + (−0.526 + 0.850i)6-s + (−0.739 + 0.673i)7-s + (−0.932 − 0.361i)8-s + (0.982 − 0.183i)9-s + i·10-s + (0.273 + 0.961i)11-s + (0.361 + 0.932i)12-s + (0.673 + 0.739i)13-s + (0.0922 + 0.995i)14-s + (0.739 − 0.673i)15-s + (−0.850 + 0.526i)16-s + (−0.932 + 0.361i)17-s + ⋯
L(s)  = 1  + (0.602 − 0.798i)2-s + (−0.995 + 0.0922i)3-s + (−0.273 − 0.961i)4-s + (−0.798 + 0.602i)5-s + (−0.526 + 0.850i)6-s + (−0.739 + 0.673i)7-s + (−0.932 − 0.361i)8-s + (0.982 − 0.183i)9-s + i·10-s + (0.273 + 0.961i)11-s + (0.361 + 0.932i)12-s + (0.673 + 0.739i)13-s + (0.0922 + 0.995i)14-s + (0.739 − 0.673i)15-s + (−0.850 + 0.526i)16-s + (−0.932 + 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.358 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4242425554 + 0.2916241824i\)
\(L(\frac12)\) \(\approx\) \(0.4242425554 + 0.2916241824i\)
\(L(1)\) \(\approx\) \(0.7048506051 - 0.04284088489i\)
\(L(1)\) \(\approx\) \(0.7048506051 - 0.04284088489i\)

Euler product

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

−28.18428861449588951228488230851, −27.22561081298159767102547210709, −26.38755092369798962012272811426, −25.00867456442623614420617386171, −24.017976952755076869199857481936, −23.45200830967322377362000918485, −22.591777386481596206940330362940, −21.74567775204308385006298585037, −20.39363828033587003829939788550, −19.18190088755325889338959405550, −17.74771853400212538176905472410, −16.88662980780234201766550367901, −15.97374160285453629186391514404, −15.52126673581861956715684539538, −13.54788121966026086997562961095, −13.02387762549547079365622978983, −11.81511531051042141918407490366, −10.89972878576891900993371483682, −9.08675289732796942571888875743, −7.80076178547607758221486442749, −6.71531642494221386851893631791, −5.73374167341099945615655500543, −4.4674718471627861818645030031, −3.51506795882738142372721228490, −0.42299776013398737073278214221, 1.95208307793027773408796043356, 3.65837089157144504964038264602, 4.56208812474426064222535966950, 6.11280942599868303922035546851, 6.82677018250908939633822911470, 8.984575125789860833017603382778, 10.29865895688140272544532370303, 11.07706724138858603281906968430, 12.23215897392189387878337268878, 12.60518495398618981892303276559, 14.32189945605433009663975124744, 15.4264973688724018505816387805, 16.173937857315136259065481770324, 17.90618561170097214885065775350, 18.718004049720838848649684471336, 19.55237304587397121256309672428, 20.83475652315625555086121907580, 22.07587390412846905105769810065, 22.53449773637129063257243602151, 23.354382570458922302868580127539, 24.21573245764510892137338414581, 25.7794120764310943897320322746, 27.13374639691748415005385396627, 28.00091408007022793360073019581, 28.681747204268212120499712434916

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