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Properties

Label 1-137-137.129-r0-0-0
Degree $1$
Conductor $137$
Sign $0.449 + 0.893i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.449 + 0.893i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.449 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.050680812 + 0.6477462349i\)
\(L(\frac12)\) \(\approx\) \(1.050680812 + 0.6477462349i\)
\(L(1)\) \(\approx\) \(1.103739747 + 0.4387916953i\)
\(L(1)\) \(\approx\) \(1.103739747 + 0.4387916953i\)

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

−28.44263018908058431170243717632, −27.525916970840375621860967759700, −26.58407519091643531505333801791, −25.61513052452389059794677728290, −24.11356053329890651655341101396, −22.77233119802382082388764303045, −22.34624547948749300594368975046, −21.13810669959412463362978730732, −20.48437231718874535938998953655, −19.79331334198679906299054531402, −18.02865014502738554475827865748, −17.32396646713927300114871923664, −16.19549037882883455905592308082, −14.68038966359152195725681997536, −13.81812043523539439274670860624, −12.89400648381929627333931506194, −11.3247871676648911022415992123, −10.65091320894134863286255941539, −9.49880853800904237328538199711, −8.96880424512111358813894607839, −6.58455977696924836743295900420, −5.186097861251758635862057969191, −4.27973322416900175920550033274, −3.066788928966200977120360290518, −1.274506003190145530913782576210, 1.750052715616100354849405751686, 3.4316143080861803070154705994, 5.59237572218287518052635844796, 6.02501802716225839707570284402, 7.04773481527938297878386188269, 8.53529008074839481621394094868, 9.23534876103424628335094918308, 11.19268329556744385529000700350, 12.46420321944944421871919418223, 13.40483252356175891068315126039, 14.1611986012249640681912392123, 15.2631012880261769895732218002, 16.66073368389530520612183334817, 17.48791880951145502966801712854, 18.42744088086596202054006658575, 19.04168990642909985868290190222, 20.95058929265420837156180508806, 22.12941978839640144827844107260, 22.643182430431965120661602393975, 24.014976640959937834205912452168, 24.76458167425045780460382503033, 25.3980469644521838339834239332, 26.20366271932351818323619872531, 27.693541472070080299514721757487, 28.73114483311955976274052363320

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