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Properties

Label 1-137-137.40-r1-0-0
Degree $1$
Conductor $137$
Sign $0.738 + 0.674i$
Analytic cond. $14.7226$
Root an. cond. $14.7226$
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  + (−0.183 − 0.982i)2-s + (0.138 − 0.990i)3-s + (−0.932 + 0.361i)4-s + (0.824 − 0.565i)5-s + (−0.998 + 0.0461i)6-s + (−0.895 + 0.445i)7-s + (0.526 + 0.850i)8-s + (−0.961 − 0.273i)9-s + (−0.707 − 0.707i)10-s + (0.361 + 0.932i)11-s + (0.228 + 0.973i)12-s + (−0.317 + 0.948i)13-s + (0.602 + 0.798i)14-s + (−0.445 − 0.895i)15-s + (0.739 − 0.673i)16-s + (−0.526 + 0.850i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.982i)2-s + (0.138 − 0.990i)3-s + (−0.932 + 0.361i)4-s + (0.824 − 0.565i)5-s + (−0.998 + 0.0461i)6-s + (−0.895 + 0.445i)7-s + (0.526 + 0.850i)8-s + (−0.961 − 0.273i)9-s + (−0.707 − 0.707i)10-s + (0.361 + 0.932i)11-s + (0.228 + 0.973i)12-s + (−0.317 + 0.948i)13-s + (0.602 + 0.798i)14-s + (−0.445 − 0.895i)15-s + (0.739 − 0.673i)16-s + (−0.526 + 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.738 + 0.674i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ 0.738 + 0.674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3806272212 + 0.1475566489i\)
\(L(\frac12)\) \(\approx\) \(0.3806272212 + 0.1475566489i\)
\(L(1)\) \(\approx\) \(0.6127731801 - 0.4063894592i\)
\(L(1)\) \(\approx\) \(0.6127731801 - 0.4063894592i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.183 - 0.982i)T \)
3 \( 1 + (0.138 - 0.990i)T \)
5 \( 1 + (0.824 - 0.565i)T \)
7 \( 1 + (-0.895 + 0.445i)T \)
11 \( 1 + (0.361 + 0.932i)T \)
13 \( 1 + (-0.317 + 0.948i)T \)
17 \( 1 + (-0.526 + 0.850i)T \)
19 \( 1 + (-0.995 - 0.0922i)T \)
23 \( 1 + (-0.998 - 0.0461i)T \)
29 \( 1 + (-0.0461 + 0.998i)T \)
31 \( 1 + (-0.638 + 0.769i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.769 - 0.638i)T \)
47 \( 1 + (-0.486 - 0.873i)T \)
53 \( 1 + (0.638 + 0.769i)T \)
59 \( 1 + (-0.273 + 0.961i)T \)
61 \( 1 + (-0.961 + 0.273i)T \)
67 \( 1 + (-0.948 - 0.317i)T \)
71 \( 1 + (0.403 + 0.914i)T \)
73 \( 1 + (0.445 - 0.895i)T \)
79 \( 1 + (-0.990 + 0.138i)T \)
83 \( 1 + (-0.973 - 0.228i)T \)
89 \( 1 + (0.565 + 0.824i)T \)
97 \( 1 + (-0.914 - 0.403i)T \)
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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

−27.72014790631387112675135566769, −26.90544267709052440770523237225, −26.08466166619958935504675031408, −25.4686650934922891702203583567, −24.41771926874937719209138414236, −22.806163567700638643093957111482, −22.406263950829305649760293313, −21.46074534044790821959121642045, −20.00092529546882456440801305192, −18.93829525294254218608613335527, −17.63926495386877982659628802800, −16.80219174050432632958029871965, −15.94263347275789145875847639498, −14.918478094028372754009709119208, −13.97044775476949529174480560892, −13.15676662068925396714028058146, −10.96211937707537601139273911228, −9.9518819678457758683774777217, −9.327102627720048495482988717088, −7.97894165050592841953808404893, −6.42420553226428370421824372530, −5.72672975762480017310795025603, −4.22821816610871212206688567786, −2.9338800279837922002232054146, −0.15954487617449820509240414629, 1.682040728310590820549004499667, 2.357969939400442248207830657394, 4.13171212968176567731485939729, 5.76711333049828718598425249539, 6.98953009987185108606492112905, 8.73084546880389233092526493688, 9.26324976579011580015033990306, 10.558588913005776902643957130794, 12.28546058477132736313928567977, 12.50977581774854493358096846520, 13.56681775385494918979072890306, 14.58809361905724301178754255276, 16.61453254192846246259297494840, 17.52882405853664138667800808262, 18.342599836436188717549587929367, 19.519219688859622388159770016488, 19.97408044125821800001889340309, 21.35025693821756783924696776082, 22.16098798907804503566541909589, 23.31040180331318217271164784544, 24.39545522995409795361852537150, 25.71764838720945065497238013301, 25.98929421328137001563085643161, 27.89914968706679105678375556708, 28.61980378295800243016038177426

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