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Properties

Label 1-137-137.38-r0-0-0
Degree $1$
Conductor $137$
Sign $-0.546 + 0.837i$
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  + (0.739 + 0.673i)2-s + (−0.850 + 0.526i)3-s + (0.0922 + 0.995i)4-s + (0.739 − 0.673i)5-s + (−0.982 − 0.183i)6-s + (−0.273 + 0.961i)7-s + (−0.602 + 0.798i)8-s + (0.445 − 0.895i)9-s + 10-s + (0.0922 + 0.995i)11-s + (−0.602 − 0.798i)12-s + (−0.273 + 0.961i)13-s + (−0.850 + 0.526i)14-s + (−0.273 + 0.961i)15-s + (−0.982 + 0.183i)16-s + (−0.602 − 0.798i)17-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)2-s + (−0.850 + 0.526i)3-s + (0.0922 + 0.995i)4-s + (0.739 − 0.673i)5-s + (−0.982 − 0.183i)6-s + (−0.273 + 0.961i)7-s + (−0.602 + 0.798i)8-s + (0.445 − 0.895i)9-s + 10-s + (0.0922 + 0.995i)11-s + (−0.602 − 0.798i)12-s + (−0.273 + 0.961i)13-s + (−0.850 + 0.526i)14-s + (−0.273 + 0.961i)15-s + (−0.982 + 0.183i)16-s + (−0.602 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.546 + 0.837i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ -0.546 + 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5954029148 + 1.099608597i\)
\(L(\frac12)\) \(\approx\) \(0.5954029148 + 1.099608597i\)
\(L(1)\) \(\approx\) \(0.9494995641 + 0.7734469517i\)
\(L(1)\) \(\approx\) \(0.9494995641 + 0.7734469517i\)

Euler product

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

−28.62439967411704688031664319583, −27.380539383630283829716556516680, −26.26657161686955144679465790491, −24.69398671910477928331572736883, −24.07309540261285157461420125772, −22.80276772888751698407662085189, −22.3810719874939171473678105165, −21.44533783287091079240870540231, −20.12815294329360451607446537390, −19.12189341212630399127784904406, −18.140535640760725074773631814925, −17.17023918931201544724530784158, −15.89113819328860597149978466957, −14.361053428453490452729188138186, −13.50989672822598691322082244, −12.79859994920745486104887920256, −11.38156788992521984822797202490, −10.63754002456176797997310124193, −9.83810948938028564237935573763, −7.57511763301943717771181812705, −6.23488248579925341601153876632, −5.692727996965958491787783532466, −4.042159453558421785499166872182, −2.60947745833656672694231198900, −1.04541522944466172834943019487, 2.3092484068559941309790031638, 4.260668343130106670718806117944, 5.14134346706495058150545616692, 6.019858217224881785845730761104, 7.118119004318374496592569160161, 9.01853101177571228290709198740, 9.68325703107481913648581741319, 11.65150858077486597977833106196, 12.23794145394172479444243352907, 13.34867290812752264761599341072, 14.62845753896584887730299959143, 15.786552808652678589112895264970, 16.36503892291853518179252519917, 17.52599649538032239567728837356, 18.15319206192827101194657732164, 20.26951648298474171686873337302, 21.22719067752820565068198302051, 22.04892615861894438532224340679, 22.64111603106689856493310789532, 23.96262523837558258399203140870, 24.66827091811135938099858630366, 25.69277157694133540328112424318, 26.6601470192235596265091478737, 28.09354606472729680800411159748, 28.718001384797871055009937582638

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