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Properties

Label 1-137-137.5-r1-0-0
Degree $1$
Conductor $137$
Sign $-0.893 + 0.449i$
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  + (−0.995 − 0.0922i)2-s + (−0.948 − 0.317i)3-s + (0.982 + 0.183i)4-s + (−0.638 + 0.769i)5-s + (0.914 + 0.403i)6-s + (0.526 + 0.850i)7-s + (−0.961 − 0.273i)8-s + (0.798 + 0.602i)9-s + (0.707 − 0.707i)10-s + (−0.183 + 0.982i)11-s + (−0.873 − 0.486i)12-s + (0.228 − 0.973i)13-s + (−0.445 − 0.895i)14-s + (0.850 − 0.526i)15-s + (0.932 + 0.361i)16-s + (0.961 − 0.273i)17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0922i)2-s + (−0.948 − 0.317i)3-s + (0.982 + 0.183i)4-s + (−0.638 + 0.769i)5-s + (0.914 + 0.403i)6-s + (0.526 + 0.850i)7-s + (−0.961 − 0.273i)8-s + (0.798 + 0.602i)9-s + (0.707 − 0.707i)10-s + (−0.183 + 0.982i)11-s + (−0.873 − 0.486i)12-s + (0.228 − 0.973i)13-s + (−0.445 − 0.895i)14-s + (0.850 − 0.526i)15-s + (0.932 + 0.361i)16-s + (0.961 − 0.273i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.893 + 0.449i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ -0.893 + 0.449i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09097303634 + 0.3835343311i\)
\(L(\frac12)\) \(\approx\) \(0.09097303634 + 0.3835343311i\)
\(L(1)\) \(\approx\) \(0.4517725203 + 0.1278950644i\)
\(L(1)\) \(\approx\) \(0.4517725203 + 0.1278950644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.995 - 0.0922i)T \)
3 \( 1 + (-0.948 - 0.317i)T \)
5 \( 1 + (-0.638 + 0.769i)T \)
7 \( 1 + (0.526 + 0.850i)T \)
11 \( 1 + (-0.183 + 0.982i)T \)
13 \( 1 + (0.228 - 0.973i)T \)
17 \( 1 + (0.961 - 0.273i)T \)
19 \( 1 + (-0.673 + 0.739i)T \)
23 \( 1 + (0.914 - 0.403i)T \)
29 \( 1 + (0.403 + 0.914i)T \)
31 \( 1 + (-0.0461 + 0.998i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.998 + 0.0461i)T \)
47 \( 1 + (-0.990 + 0.138i)T \)
53 \( 1 + (0.0461 + 0.998i)T \)
59 \( 1 + (-0.602 + 0.798i)T \)
61 \( 1 + (0.798 - 0.602i)T \)
67 \( 1 + (-0.973 - 0.228i)T \)
71 \( 1 + (0.565 - 0.824i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (0.317 + 0.948i)T \)
83 \( 1 + (-0.486 - 0.873i)T \)
89 \( 1 + (0.769 + 0.638i)T \)
97 \( 1 + (-0.824 + 0.565i)T \)
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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

−27.798185207927269008617239632491, −26.99161529957503155629689850282, −26.31223618961918902578947222910, −24.683373380977648940913099479062, −23.73948468551248776431572029281, −23.398314303070690048856393961341, −21.35749332620699306522208649193, −20.93315710217355748016658476403, −19.549515695373116447999592796171, −18.71877574695886965689230297105, −17.33526830702161871822117088098, −16.744665767576410060567614498961, −16.082738947009474867455421350600, −14.89393774851700306813195060441, −13.13305960978346090003749829349, −11.586008054859631229725006695423, −11.25584232762355564867645451428, −10.01243110266471252769001310012, −8.768099345863719652419119859737, −7.684777148758565141128685366268, −6.48763065201957152181700885462, −5.1191925745548827288748977880, −3.79268529003120959101839492956, −1.30053055590094922997067519658, −0.27815214646997747062922401816, 1.49853546695573573722380204072, 3.00776323279956830393876462112, 5.10500956114962168570630524073, 6.438653276054479169163613996641, 7.45719630614977978162072493576, 8.37738897209517164352097439196, 10.14485532615406161086233099876, 10.804294537030852953552523352927, 11.970948019009924292988314402, 12.504529774522010997854746236104, 14.81531329927274916809973195452, 15.51949467121925377099006452670, 16.67879934642405771500853828890, 17.85914824629071967742283058643, 18.38170572950406986526630166003, 19.17647200666596375379216263208, 20.53173575125686790228675374805, 21.649674086742911138027939929326, 22.84017323882488665171757622852, 23.62558395427636613599780068527, 24.99856650090993008575965736406, 25.5692097622750728204787473986, 27.25098120514700037065217156890, 27.528886061036615222893429660010, 28.44345861286893578679359758724

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