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Properties

Label 1-4031-4031.1235-r0-0-0
Degree $1$
Conductor $4031$
Sign $-0.931 - 0.363i$
Analytic cond. $18.7198$
Root an. cond. $18.7198$
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  + (−0.181 + 0.983i)2-s + (−0.926 + 0.377i)3-s + (−0.934 − 0.356i)4-s + (0.998 − 0.0455i)5-s + (−0.203 − 0.979i)6-s + (−0.949 + 0.313i)7-s + (0.519 − 0.854i)8-s + (0.715 − 0.699i)9-s + (−0.136 + 0.990i)10-s + (0.956 − 0.291i)11-s + (0.999 − 0.0227i)12-s + (−0.613 + 0.789i)13-s + (−0.136 − 0.990i)14-s + (−0.907 + 0.419i)15-s + (0.746 + 0.665i)16-s + (−0.595 + 0.803i)17-s + ⋯
L(s)  = 1  + (−0.181 + 0.983i)2-s + (−0.926 + 0.377i)3-s + (−0.934 − 0.356i)4-s + (0.998 − 0.0455i)5-s + (−0.203 − 0.979i)6-s + (−0.949 + 0.313i)7-s + (0.519 − 0.854i)8-s + (0.715 − 0.699i)9-s + (−0.136 + 0.990i)10-s + (0.956 − 0.291i)11-s + (0.999 − 0.0227i)12-s + (−0.613 + 0.789i)13-s + (−0.136 − 0.990i)14-s + (−0.907 + 0.419i)15-s + (0.746 + 0.665i)16-s + (−0.595 + 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $-0.931 - 0.363i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (1235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ -0.931 - 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1462794050 + 0.7773269596i\)
\(L(\frac12)\) \(\approx\) \(-0.1462794050 + 0.7773269596i\)
\(L(1)\) \(\approx\) \(0.5265733007 + 0.4796920449i\)
\(L(1)\) \(\approx\) \(0.5265733007 + 0.4796920449i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.181 + 0.983i)T \)
3 \( 1 + (-0.926 + 0.377i)T \)
5 \( 1 + (0.998 - 0.0455i)T \)
7 \( 1 + (-0.949 + 0.313i)T \)
11 \( 1 + (0.956 - 0.291i)T \)
13 \( 1 + (-0.613 + 0.789i)T \)
17 \( 1 + (-0.595 + 0.803i)T \)
19 \( 1 + (0.993 + 0.113i)T \)
23 \( 1 + (-0.203 + 0.979i)T \)
31 \( 1 + (-0.699 + 0.715i)T \)
37 \( 1 + (-0.907 - 0.419i)T \)
41 \( 1 + (0.439 + 0.898i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.842 + 0.538i)T \)
53 \( 1 + (0.648 - 0.761i)T \)
59 \( 1 + (0.775 + 0.631i)T \)
61 \( 1 + (-0.439 + 0.898i)T \)
67 \( 1 + (0.974 - 0.225i)T \)
71 \( 1 + (-0.538 - 0.842i)T \)
73 \( 1 + (0.480 + 0.877i)T \)
79 \( 1 + (-0.997 + 0.0682i)T \)
83 \( 1 + (-0.974 - 0.225i)T \)
89 \( 1 + (0.0909 - 0.995i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−18.159212658386125243945004843212, −17.46577053605442666488298218431, −16.99117699263666826728704790063, −16.45956748015365445885943562516, −15.48480819526831662350474294274, −14.28888556017136985399887558853, −13.765211298511136054168258720442, −13.08697899343445329916824973928, −12.54862302663274371275761262131, −11.983854573854579232191418098790, −11.19763335525620429842469923599, −10.44787520522230655501697145052, −9.87722337673758763081165310945, −9.46338193798513404732519335687, −8.609130594228511871362186003452, −7.33200912330761612225627004189, −6.92166578984661479813287518595, −5.988279914656206397637102470670, −5.29777725236737153863812686704, −4.5985568868775240106031643617, −3.658301795566085697195332771592, −2.69829573084615780766664435352, −2.05463628075410328203716781011, −1.07281332284574493810046019410, −0.337983069935570044445393787624, 1.06487304060115656852586825561, 1.86450004577992184203479014844, 3.364326856331086814346096850586, 4.052017088262714812449901449274, 4.96824680275382456666678299812, 5.7016863211407725073145757602, 6.06699712864420487249894915898, 6.87415012732833029397795003793, 7.18789237732522906051537814767, 8.7824218215019109661737746239, 9.10686560220836928670639304235, 9.84926031639411700754575681341, 10.177005209345339349120462178943, 11.25504359646649777828358581237, 12.09450359242449698080667909184, 12.78246299433819903092586497193, 13.4894022949975084090791601287, 14.18828232096878120413371052023, 14.86046773372637840995723719023, 15.694572599612846528831035858654, 16.257219630802685508225829685365, 16.84460260215492553884038308214, 17.27687458191919948852928520185, 17.9474489169699201771339303857, 18.53162887269545797646527311695

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