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Properties

Label 1-4031-4031.576-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  + (0.254 + 0.967i)2-s + (0.999 + 0.0130i)3-s + (−0.870 + 0.491i)4-s + (−0.851 + 0.525i)5-s + (0.241 + 0.970i)6-s + (−0.0552 − 0.998i)7-s + (−0.696 − 0.717i)8-s + (0.999 + 0.0260i)9-s + (−0.724 − 0.689i)10-s + (−0.00325 − 0.999i)11-s + (−0.877 + 0.480i)12-s + (0.818 + 0.574i)13-s + (0.951 − 0.307i)14-s + (−0.857 + 0.514i)15-s + (0.516 − 0.856i)16-s + (0.113 + 0.993i)17-s + ⋯
L(s)  = 1  + (0.254 + 0.967i)2-s + (0.999 + 0.0130i)3-s + (−0.870 + 0.491i)4-s + (−0.851 + 0.525i)5-s + (0.241 + 0.970i)6-s + (−0.0552 − 0.998i)7-s + (−0.696 − 0.717i)8-s + (0.999 + 0.0260i)9-s + (−0.724 − 0.689i)10-s + (−0.00325 − 0.999i)11-s + (−0.877 + 0.480i)12-s + (0.818 + 0.574i)13-s + (0.951 − 0.307i)14-s + (−0.857 + 0.514i)15-s + (0.516 − 0.856i)16-s + (0.113 + 0.993i)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: $\chi_{4031} (576, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.931 + 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.113298243 + 0.3971613097i\)
\(L(\frac12)\) \(\approx\) \(2.113298243 + 0.3971613097i\)
\(L(1)\) \(\approx\) \(1.274053647 + 0.5132096571i\)
\(L(1)\) \(\approx\) \(1.274053647 + 0.5132096571i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.254 + 0.967i)T \)
3 \( 1 + (0.999 + 0.0130i)T \)
5 \( 1 + (-0.851 + 0.525i)T \)
7 \( 1 + (-0.0552 - 0.998i)T \)
11 \( 1 + (-0.00325 - 0.999i)T \)
13 \( 1 + (0.818 + 0.574i)T \)
17 \( 1 + (0.113 + 0.993i)T \)
19 \( 1 + (0.966 + 0.257i)T \)
23 \( 1 + (-0.608 - 0.793i)T \)
31 \( 1 + (-0.197 + 0.980i)T \)
37 \( 1 + (-0.132 - 0.991i)T \)
41 \( 1 + (-0.829 + 0.557i)T \)
43 \( 1 + (0.0747 - 0.997i)T \)
47 \( 1 + (0.177 - 0.984i)T \)
53 \( 1 + (-0.964 - 0.263i)T \)
59 \( 1 + (-0.775 + 0.631i)T \)
61 \( 1 + (-0.953 + 0.300i)T \)
67 \( 1 + (0.867 - 0.497i)T \)
71 \( 1 + (0.266 - 0.963i)T \)
73 \( 1 + (-0.454 - 0.890i)T \)
79 \( 1 + (0.987 - 0.155i)T \)
83 \( 1 + (0.929 - 0.368i)T \)
89 \( 1 + (0.771 - 0.636i)T \)
97 \( 1 + (0.365 - 0.930i)T \)
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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

−18.56359181430029545036658235535, −18.195493977821916616518507321973, −17.314922266605010919847265035762, −15.86748893685123375772710874758, −15.67942732013609637031059138456, −15.03028383783426903478544314924, −14.25741896804959842717857593194, −13.47420186730509789170888011505, −12.925264752241760991680605061371, −12.23903532708924101595352417113, −11.74635813409328474683292259304, −11.04032555986502659597744780169, −9.88632579933522512531840425227, −9.4338990680351935311872189446, −8.93720500266158131846929845363, −7.97796607801342422105616259576, −7.70330487108662256451871417953, −6.42761518837200895237876068113, −5.27699677102382052806763621876, −4.821081773574316306380764632362, −3.9206996077584873111084170716, −3.261125539575014318323485257989, −2.66059316740736435934678619139, −1.74073782261467757636423956050, −0.99525924683956266062624863238, 0.567177027488784809398399957327, 1.70585327334136951324008342411, 3.25135732873510929347425252161, 3.484914013479296079854348484407, 4.07074517531054909023376368, 4.86336193371105160260029192276, 6.13208296078362132154128691454, 6.64863169829171558569770193866, 7.44696075483228764237321862070, 7.93456035148601074840325509418, 8.56341704545920252410589294614, 9.13322759448124136982676264609, 10.291129996559872428886817759225, 10.68647079701004929346187549148, 11.798138051943302503549706790101, 12.57468929094629143358767367465, 13.416876776175009105893062994669, 14.06760970787370090665624578762, 14.212896472023834106993785050517, 15.1198316900495239662388253300, 15.76446305201156852132726860745, 16.36392552170894443818581033444, 16.745098428401394837957239401, 18.01150704070855120004266535610, 18.4637781232623064128252088613

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