↑

Properties

Label 1-137-137.48-r1-0-0
Degree $1$
Conductor $137$
Sign $0.000495 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.995 + 0.0922i)2-s + (−0.317 + 0.948i)3-s + (0.982 + 0.183i)4-s + (−0.769 − 0.638i)5-s + (−0.403 + 0.914i)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 + (−0.486 + 0.873i)12-s + (−0.973 − 0.228i)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 + (−0.317 + 0.948i)3-s + (0.982 + 0.183i)4-s + (−0.769 − 0.638i)5-s + (−0.403 + 0.914i)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 + (−0.486 + 0.873i)12-s + (−0.973 − 0.228i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.000495 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.000495 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.000495 - 0.999i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ 0.000495 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.044894965 - 1.045413125i\)
\(L(\frac12)\) \(\approx\) \(1.044894965 - 1.045413125i\)
\(L(1)\) \(\approx\) \(1.253793253 - 0.09168214508i\)
\(L(1)\) \(\approx\) \(1.253793253 - 0.09168214508i\)

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

Imaginary part of the first few zeros on the critical line

−28.822259279075285978175394368506, −27.84371275089195796503999082524, −26.21431254660732304233239002408, −25.13574552006723785807493644576, −24.414073644418852131454206684080, −23.381186799064246484773667499038, −22.46339825063272898616848392293, −22.1262692158513290991251168033, −20.307112750637618278373318212362, −19.45688347886836797700197534781, −18.678028520788513681178555054994, −17.37448572670957089738662865339, −15.92648188568410964060659567572, −15.07611822807520153748507314672, −14.052865847050954904434293698066, −12.79458806014518611892309394826, −11.99034106388280597503545929429, −11.41433203466273575088071740730, −9.810231556019406445756623470, −7.78877556986275789448359980934, −6.969664356179274624277831113591, −6.00202117835878597456713528944, −4.6321117246847163539750986981, −3.02438581841519720676153606776, −1.98794359008904868086228420487, 0.406173957573224235569462554611, 3.06247634093443469091567919894, 4.104354811203795504617208190351, 4.896959737335624222682935463593, 6.22923404265218701672264300150, 7.56117001879894148589572094675, 9.04146120263391838858961416731, 10.536511703067955324541343309342, 11.37648116141142855784988498989, 12.441752586913357383310665278540, 13.5845855267132810082788884204, 14.75874969554398252406244451330, 15.830503898750607589048172342557, 16.42434976643767913027960925048, 17.2688890978573052446197500199, 19.62722784788937685016396374840, 20.03144858939007869182443967954, 21.12465801976079011187365260944, 22.23239794740280078319929694958, 22.78797452506347397094890033706, 23.9771321616492900808050066118, 24.55343621584962932661439348440, 26.34005563397294334889986904830, 26.74768998793816994277822244501, 28.15517285205946452362989581856

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