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Properties

Label 1-139-139.71-r0-0-0
Degree $1$
Conductor $139$
Sign $0.995 - 0.0950i$
Analytic cond. $0.645513$
Root an. cond. $0.645513$
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  + (−0.247 + 0.968i)2-s + (0.746 − 0.665i)3-s + (−0.877 − 0.480i)4-s + (0.898 − 0.439i)5-s + (0.460 + 0.887i)6-s + (−0.998 + 0.0455i)7-s + (0.682 − 0.730i)8-s + (0.113 − 0.993i)9-s + (0.203 + 0.979i)10-s + (0.983 + 0.181i)11-s + (−0.974 + 0.225i)12-s + (−0.949 − 0.313i)13-s + (0.203 − 0.979i)14-s + (0.377 − 0.926i)15-s + (0.538 + 0.842i)16-s + (0.995 + 0.0909i)17-s + ⋯
L(s)  = 1  + (−0.247 + 0.968i)2-s + (0.746 − 0.665i)3-s + (−0.877 − 0.480i)4-s + (0.898 − 0.439i)5-s + (0.460 + 0.887i)6-s + (−0.998 + 0.0455i)7-s + (0.682 − 0.730i)8-s + (0.113 − 0.993i)9-s + (0.203 + 0.979i)10-s + (0.983 + 0.181i)11-s + (−0.974 + 0.225i)12-s + (−0.949 − 0.313i)13-s + (0.203 − 0.979i)14-s + (0.377 − 0.926i)15-s + (0.538 + 0.842i)16-s + (0.995 + 0.0909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.995 - 0.0950i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.995 - 0.0950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.186229830 - 0.05648350646i\)
\(L(\frac12)\) \(\approx\) \(1.186229830 - 0.05648350646i\)
\(L(1)\) \(\approx\) \(1.137509412 + 0.07412306595i\)
\(L(1)\) \(\approx\) \(1.137509412 + 0.07412306595i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (-0.247 + 0.968i)T \)
3 \( 1 + (0.746 - 0.665i)T \)
5 \( 1 + (0.898 - 0.439i)T \)
7 \( 1 + (-0.998 + 0.0455i)T \)
11 \( 1 + (0.983 + 0.181i)T \)
13 \( 1 + (-0.949 - 0.313i)T \)
17 \( 1 + (0.995 + 0.0909i)T \)
19 \( 1 + (-0.419 - 0.907i)T \)
23 \( 1 + (0.460 - 0.887i)T \)
29 \( 1 + (0.0227 + 0.999i)T \)
31 \( 1 + (0.113 + 0.993i)T \)
37 \( 1 + (0.377 + 0.926i)T \)
41 \( 1 + (-0.158 + 0.987i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.829 + 0.557i)T \)
53 \( 1 + (-0.715 - 0.699i)T \)
59 \( 1 + (0.854 + 0.519i)T \)
61 \( 1 + (-0.158 - 0.987i)T \)
67 \( 1 + (-0.648 - 0.761i)T \)
71 \( 1 + (-0.829 - 0.557i)T \)
73 \( 1 + (0.291 + 0.956i)T \)
79 \( 1 + (-0.775 + 0.631i)T \)
83 \( 1 + (-0.648 + 0.761i)T \)
89 \( 1 + (0.613 + 0.789i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−28.49292216879696533426439197745, −27.35988294647634929391454413340, −26.61311143414335921607859156395, −25.66719896987044391617377352477, −25.00929750688739040105803583447, −22.90794079078703766381438647537, −22.11507504612898557316387008336, −21.43303884846626174231168815992, −20.50091658753218267643120931480, −19.24964399449094718494712286921, −18.98315107509580129169174371743, −17.23519693221895377478822760526, −16.582037298340306407647461813289, −14.80791132917487967288172013895, −14.00386096459373695976712756046, −13.07112218807945640925497655315, −11.770047649454933862652008312599, −10.275778141483443642594696858601, −9.74063047412908921122867363550, −9.00015339284154045084010009737, −7.41570114649430624251557102717, −5.67885936143033388425433424710, −4.025002176546367770882585426650, −3.073193721130629801125678244221, −1.92138772849726416247041538129, 1.239533845113072159012349296555, 3.03120269732982723769538875577, 4.82993126560900634211120262481, 6.330998322673946996712620579444, 6.92360375235819388104917332835, 8.38159241640027334736462823075, 9.338911046740471718820585432875, 9.97622664532716811926865740291, 12.48268105348811532883105041780, 13.08051917937219856006800290698, 14.2455324521236696610051278376, 14.90201578095540071072686946437, 16.41081598528337982460310481352, 17.21510147379142610171151210481, 18.20154764862733027015074972413, 19.32824517737235245247665100768, 19.99557517356049573211504196843, 21.60346471018015045236072916648, 22.61659606581069251131214260863, 23.81511551212951521922763329718, 24.809744852998597552850030095250, 25.336872301398974020910600163790, 26.01468459240488321885732671336, 27.14385336932253449139635308031, 28.404545181528214623063833150662

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