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Properties

Label 1-137-137.60-r0-0-0
Degree $1$
Conductor $137$
Sign $0.893 - 0.449i$
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  + (0.932 + 0.361i)2-s + (−0.273 − 0.961i)3-s + (0.739 + 0.673i)4-s + (0.932 − 0.361i)5-s + (0.0922 − 0.995i)6-s + (−0.602 − 0.798i)7-s + (0.445 + 0.895i)8-s + (−0.850 + 0.526i)9-s + 10-s + (0.739 + 0.673i)11-s + (0.445 − 0.895i)12-s + (−0.602 − 0.798i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 + 0.995i)16-s + (0.445 − 0.895i)17-s + ⋯
L(s)  = 1  + (0.932 + 0.361i)2-s + (−0.273 − 0.961i)3-s + (0.739 + 0.673i)4-s + (0.932 − 0.361i)5-s + (0.0922 − 0.995i)6-s + (−0.602 − 0.798i)7-s + (0.445 + 0.895i)8-s + (−0.850 + 0.526i)9-s + 10-s + (0.739 + 0.673i)11-s + (0.445 − 0.895i)12-s + (−0.602 − 0.798i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 + 0.995i)16-s + (0.445 − 0.895i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.893 - 0.449i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.893 - 0.449i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.761015274 - 0.4180671300i\)
\(L(\frac12)\) \(\approx\) \(1.761015274 - 0.4180671300i\)
\(L(1)\) \(\approx\) \(1.662869270 - 0.2192396311i\)
\(L(1)\) \(\approx\) \(1.662869270 - 0.2192396311i\)

Euler product

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

−28.72173129642195744536483983980, −27.92171154363716036401761564237, −26.49525691156128202447228437865, −25.51785719058774466786004683396, −24.582139860855373460527025719703, −23.25488063228327479377783193909, −22.15424797814575691748718044983, −21.73266473163125074216468583671, −21.118224560301098845260893547877, −19.665610954637837795348733671994, −18.75093942248323830220836066128, −17.03819059258486968559308306870, −16.29167988784645026588206928852, −14.85950941895941351001344108257, −14.485994037661833183290949508214, −13.0687568809214339440310474340, −11.94764166665047900550722929803, −10.87864320680665646606545301212, −9.88666114022045935686414173943, −8.98669875028300719447755579041, −6.38690741625807617077709714536, −5.99310722195540481591190956840, −4.62792508676711985681225373795, −3.36375080481735464433097404062, −2.174072202833446216222930707013, 1.599684479207998220852906993045, 3.02732869644050841292628539455, 4.76373164658700788512710699492, 5.90248369115246375553665305422, 6.848484099093510236241508493881, 7.7413092137649194912609603042, 9.49541273578983878677394679247, 10.93844293023092049979995871048, 12.39257128183341555366044837761, 12.91041261879748918746429374346, 13.86154479527307357777072901176, 14.72701680390325973285907010765, 16.46971804084406178408637966533, 17.09773947259925077090170707498, 17.95969743354833849467388863995, 19.71348264487059482201142555853, 20.31258433138023563648880478680, 21.71646893860506978750918775863, 22.68793516508085806026521002692, 23.36308147594867446184785403704, 24.45183904649463612875891758263, 25.3841635575704571971305759606, 25.67705559577029953152894153273, 27.512296223382878638566940678359, 28.96780438806957960185592634708

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