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Properties

Label 1-137-137.120-r0-0-0
Degree $1$
Conductor $137$
Sign $-0.898 + 0.439i$
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  + (0.273 − 0.961i)2-s + (0.183 − 0.982i)3-s + (−0.850 − 0.526i)4-s + (−0.961 + 0.273i)5-s + (−0.895 − 0.445i)6-s + (−0.0922 − 0.995i)7-s + (−0.739 + 0.673i)8-s + (−0.932 − 0.361i)9-s + i·10-s + (0.850 + 0.526i)11-s + (−0.673 + 0.739i)12-s + (−0.995 + 0.0922i)13-s + (−0.982 − 0.183i)14-s + (0.0922 + 0.995i)15-s + (0.445 + 0.895i)16-s + (−0.739 − 0.673i)17-s + ⋯
L(s)  = 1  + (0.273 − 0.961i)2-s + (0.183 − 0.982i)3-s + (−0.850 − 0.526i)4-s + (−0.961 + 0.273i)5-s + (−0.895 − 0.445i)6-s + (−0.0922 − 0.995i)7-s + (−0.739 + 0.673i)8-s + (−0.932 − 0.361i)9-s + i·10-s + (0.850 + 0.526i)11-s + (−0.673 + 0.739i)12-s + (−0.995 + 0.0922i)13-s + (−0.982 − 0.183i)14-s + (0.0922 + 0.995i)15-s + (0.445 + 0.895i)16-s + (−0.739 − 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.898 + 0.439i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (120, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ -0.898 + 0.439i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1692313395 - 0.7307380922i\)
\(L(\frac12)\) \(\approx\) \(-0.1692313395 - 0.7307380922i\)
\(L(1)\) \(\approx\) \(0.4529300230 - 0.7277431858i\)
\(L(1)\) \(\approx\) \(0.4529300230 - 0.7277431858i\)

Euler product

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

−28.65612979914801940870058186650, −27.64433833385412002715463184756, −27.033801884139014462886594644443, −26.21964342583531895088812391049, −24.89209526222757416343650789101, −24.37743167425519835884094875375, −22.957681682378616851700255705136, −22.15321971535304235830565705107, −21.49540147284140012520104500294, −19.98792771475517407961835216658, −19.06635314059016688768258142825, −17.58033613967286944564325326731, −16.44303264699588233425540954072, −15.81756475765588136510926390423, −14.91871975644595922702101283788, −14.17864811663744010572722189666, −12.44876216902440230429350379051, −11.66228504510659380233097760961, −9.90169561307910404433228830252, −8.72588365684851876972907984890, −8.1600769532600335217937033756, −6.47286892059297183413390081309, −5.21082405872141706540075609928, −4.22174505657767835461159804401, −3.11852187625920370924806484986, 0.60648496489733105278161590854, 2.27030889183531427670450779535, 3.59340116234635353419375094659, 4.72044159954461745406916685307, 6.711148477014540012358441186487, 7.56994755931491795736793732668, 8.9866805061076397013290425131, 10.295633221844805359906765654317, 11.74322099958396545067058018974, 12.00178081610014103591510109615, 13.47549054589497843791914406271, 14.136685068752079489969818799188, 15.315264026136443352088911607140, 17.1670956215324843460815495686, 17.99791818646696435465196524936, 19.29792009125864188763901938515, 19.79775378477091606631395933908, 20.39537409909623008211593371589, 22.2386715287000280596692211990, 22.84763808100783846771925602209, 23.81128432678199264955980435196, 24.551245847625496343659499905883, 26.26176308645225541669990328390, 26.99160024923613675495133439393, 28.09127479025539181400271243892

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