Properties

Label 1-4031-4031.6-r0-0-0
Degree $1$
Conductor $4031$
Sign $-0.894 - 0.446i$
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  + (−0.993 − 0.116i)2-s + (−0.951 − 0.307i)3-s + (0.972 + 0.232i)4-s + (0.763 − 0.646i)5-s + (0.909 + 0.416i)6-s + (0.241 − 0.970i)7-s + (−0.938 − 0.344i)8-s + (0.811 + 0.584i)9-s + (−0.833 + 0.552i)10-s + (−0.996 + 0.0779i)11-s + (−0.854 − 0.519i)12-s + (−0.511 + 0.859i)13-s + (−0.353 + 0.935i)14-s + (−0.924 + 0.380i)15-s + (0.892 + 0.451i)16-s + (0.917 + 0.398i)17-s + ⋯
L(s)  = 1  + (−0.993 − 0.116i)2-s + (−0.951 − 0.307i)3-s + (0.972 + 0.232i)4-s + (0.763 − 0.646i)5-s + (0.909 + 0.416i)6-s + (0.241 − 0.970i)7-s + (−0.938 − 0.344i)8-s + (0.811 + 0.584i)9-s + (−0.833 + 0.552i)10-s + (−0.996 + 0.0779i)11-s + (−0.854 − 0.519i)12-s + (−0.511 + 0.859i)13-s + (−0.353 + 0.935i)14-s + (−0.924 + 0.380i)15-s + (0.892 + 0.451i)16-s + (0.917 + 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $-0.894 - 0.446i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ -0.894 - 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1016449056 - 0.4310184799i\)
\(L(\frac12)\) \(\approx\) \(0.1016449056 - 0.4310184799i\)
\(L(1)\) \(\approx\) \(0.4968074589 - 0.1754637899i\)
\(L(1)\) \(\approx\) \(0.4968074589 - 0.1754637899i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.993 - 0.116i)T \)
3 \( 1 + (-0.951 - 0.307i)T \)
5 \( 1 + (0.763 - 0.646i)T \)
7 \( 1 + (0.241 - 0.970i)T \)
11 \( 1 + (-0.996 + 0.0779i)T \)
13 \( 1 + (-0.511 + 0.859i)T \)
17 \( 1 + (0.917 + 0.398i)T \)
19 \( 1 + (-0.999 + 0.0390i)T \)
23 \( 1 + (-0.999 - 0.0195i)T \)
31 \( 1 + (-0.0487 + 0.998i)T \)
37 \( 1 + (0.998 - 0.0585i)T \)
41 \( 1 + (0.0682 + 0.997i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (0.407 + 0.913i)T \)
53 \( 1 + (0.993 + 0.116i)T \)
59 \( 1 + (-0.775 + 0.631i)T \)
61 \( 1 + (-0.494 + 0.869i)T \)
67 \( 1 + (0.996 + 0.0779i)T \)
71 \( 1 + (0.981 + 0.193i)T \)
73 \( 1 + (-0.316 - 0.948i)T \)
79 \( 1 + (0.822 - 0.568i)T \)
83 \( 1 + (-0.932 - 0.362i)T \)
89 \( 1 + (0.668 - 0.744i)T \)
97 \( 1 + (0.900 - 0.433i)T \)
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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

−18.448929459910436281127615504235, −18.19574374418569421930365689554, −17.427197741765198065489976092837, −16.95608597823977106885484714953, −16.14716642678123612536968655076, −15.408756703044451375462378992227, −15.04948128131376073817910213589, −14.27652190870639951437693100449, −13.031563635408669472122977636403, −12.4110719016250964498423589297, −11.719277523484354127008248148324, −10.91498636167249115734775069878, −10.49178114377178996126372075970, −9.75582841071071397056191682952, −9.41558621483150567046128276673, −8.19116256059409874692640003988, −7.71328738152286818929996809466, −6.773406605323776027238764121423, −5.979573712574118619465155925248, −5.61636084370404522331785453891, −4.99629546341050527975183079976, −3.557867145889532883117210276939, −2.464227559877947435115606507873, −2.19185915155278488544328847515, −0.88696019730024324488642344218, 0.24109674716948790210738789746, 1.23640111458338308630661175198, 1.80337284105854415394309032721, 2.61335796410570680561428219861, 4.047838393511279061811432188544, 4.737322515433098376188893707579, 5.66986386190250638084610637599, 6.25312371198941616438051277106, 7.04507016743508060542800600915, 7.733104544326029477831636848106, 8.30422985694887493544725050292, 9.326861960715577676039111308744, 10.125549315362812299796120272122, 10.399584388934073581898954622195, 11.078164570133250846044627982634, 12.04623386686800729044027578208, 12.48313912656848502171543279267, 13.20436457875437559690489473942, 13.98280579977983385016756929210, 14.8646768022903022916957392434, 15.969502864248508685877958367385, 16.48667097242672716907195055505, 16.91110087087882768953179496388, 17.44327214661031855863707917756, 18.073754798036250519083279188840

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