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Properties

Label 1-137-137.12-r1-0-0
Degree $1$
Conductor $137$
Sign $0.960 + 0.277i$
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  + (−0.961 − 0.273i)2-s + (0.565 + 0.824i)3-s + (0.850 + 0.526i)4-s + (−0.873 − 0.486i)5-s + (−0.317 − 0.948i)6-s + (−0.995 + 0.0922i)7-s + (−0.673 − 0.739i)8-s + (−0.361 + 0.932i)9-s + (0.707 + 0.707i)10-s + (0.526 − 0.850i)11-s + (0.0461 + 0.998i)12-s + (0.638 − 0.769i)13-s + (0.982 + 0.183i)14-s + (−0.0922 − 0.995i)15-s + (0.445 + 0.895i)16-s + (0.673 − 0.739i)17-s + ⋯
L(s)  = 1  + (−0.961 − 0.273i)2-s + (0.565 + 0.824i)3-s + (0.850 + 0.526i)4-s + (−0.873 − 0.486i)5-s + (−0.317 − 0.948i)6-s + (−0.995 + 0.0922i)7-s + (−0.673 − 0.739i)8-s + (−0.361 + 0.932i)9-s + (0.707 + 0.707i)10-s + (0.526 − 0.850i)11-s + (0.0461 + 0.998i)12-s + (0.638 − 0.769i)13-s + (0.982 + 0.183i)14-s + (−0.0922 − 0.995i)15-s + (0.445 + 0.895i)16-s + (0.673 − 0.739i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ 0.960 + 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.049338383 + 0.1483751295i\)
\(L(\frac12)\) \(\approx\) \(1.049338383 + 0.1483751295i\)
\(L(1)\) \(\approx\) \(0.7533743813 + 0.05848807031i\)
\(L(1)\) \(\approx\) \(0.7533743813 + 0.05848807031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.961 - 0.273i)T \)
3 \( 1 + (0.565 + 0.824i)T \)
5 \( 1 + (-0.873 - 0.486i)T \)
7 \( 1 + (-0.995 + 0.0922i)T \)
11 \( 1 + (0.526 - 0.850i)T \)
13 \( 1 + (0.638 - 0.769i)T \)
17 \( 1 + (0.673 - 0.739i)T \)
19 \( 1 + (0.798 + 0.602i)T \)
23 \( 1 + (-0.317 + 0.948i)T \)
29 \( 1 + (0.948 + 0.317i)T \)
31 \( 1 + (-0.138 + 0.990i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (0.990 - 0.138i)T \)
47 \( 1 + (0.914 - 0.403i)T \)
53 \( 1 + (0.138 + 0.990i)T \)
59 \( 1 + (0.932 + 0.361i)T \)
61 \( 1 + (-0.361 - 0.932i)T \)
67 \( 1 + (0.769 + 0.638i)T \)
71 \( 1 + (0.973 + 0.228i)T \)
73 \( 1 + (0.0922 - 0.995i)T \)
79 \( 1 + (0.824 + 0.565i)T \)
83 \( 1 + (-0.998 - 0.0461i)T \)
89 \( 1 + (0.486 - 0.873i)T \)
97 \( 1 + (-0.228 - 0.973i)T \)
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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

−28.24827843631606875652531664647, −26.99296882197966176294623953818, −25.88811660640637933097615214264, −25.75136635922899131318970673811, −24.310846825340545210840373416836, −23.53447620678081046320631740659, −22.534524703984653940361768829104, −20.602986694383922401742772722273, −19.769427459919639225792108987661, −19.056141420074693703581436992478, −18.40662855126274795050821390115, −17.151472430312819989659873347414, −15.96568886296301819426414155016, −15.042531160493676277693629356063, −14.00832185236280747780029856243, −12.430671151632333022176628493606, −11.60762688546939873707780786486, −10.13117329848592698025402945006, −9.04901411937714319977573276917, −7.96566573967050330390682814867, −6.939694331870351523566479248345, −6.339861709308862494406817763193, −3.76046709224074745177554761441, −2.439802694037618550188408544197, −0.816705113448601973177641493904, 0.85310230539545489573800303633, 3.14016193138177860565655027206, 3.64564869525792165852596873306, 5.66187905621601930511729345741, 7.41072951442967591186858767961, 8.45336902326099727017175634701, 9.25694092027515329573845559007, 10.27480443196465881534082643336, 11.418180778040440406624104262471, 12.4835497658399142878776058367, 13.94845563707220704743954878181, 15.63502534311592172781682552181, 16.00570134955949979780919539503, 16.80247914358742246535015417106, 18.454485095769013298922370801615, 19.50173083860219582631639795598, 19.98701991772989439692823171132, 20.9694824957205922149685527535, 22.05493035198435529080017128529, 23.223890726121247064124602396204, 24.8967540050142672045660922357, 25.39458868736313765376929656143, 26.643421499423030658419859298477, 27.25064972473564081494966939068, 27.99135902490560061769474951087

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