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Properties

Label 1-4031-4031.31-r1-0-0
Degree $1$
Conductor $4031$
Sign $0.779 - 0.625i$
Analytic cond. $433.191$
Root an. cond. $433.191$
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  + (−0.933 + 0.359i)2-s + (0.404 − 0.914i)3-s + (0.741 − 0.670i)4-s + (−0.401 − 0.915i)5-s + (−0.0487 + 0.998i)6-s + (−0.197 − 0.980i)7-s + (−0.451 + 0.892i)8-s + (−0.672 − 0.739i)9-s + (0.703 + 0.710i)10-s + (−0.103 − 0.994i)11-s + (−0.313 − 0.949i)12-s + (−0.754 − 0.655i)13-s + (0.536 + 0.844i)14-s + (−0.999 − 0.00325i)15-s + (0.100 − 0.994i)16-s + (0.480 + 0.877i)17-s + ⋯
L(s)  = 1  + (−0.933 + 0.359i)2-s + (0.404 − 0.914i)3-s + (0.741 − 0.670i)4-s + (−0.401 − 0.915i)5-s + (−0.0487 + 0.998i)6-s + (−0.197 − 0.980i)7-s + (−0.451 + 0.892i)8-s + (−0.672 − 0.739i)9-s + (0.703 + 0.710i)10-s + (−0.103 − 0.994i)11-s + (−0.313 − 0.949i)12-s + (−0.754 − 0.655i)13-s + (0.536 + 0.844i)14-s + (−0.999 − 0.00325i)15-s + (0.100 − 0.994i)16-s + (0.480 + 0.877i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.779 - 0.625i$
Analytic conductor: \(433.191\)
Root analytic conductor: \(433.191\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (1:\ ),\ 0.779 - 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7695055543 - 0.2706246407i\)
\(L(\frac12)\) \(\approx\) \(0.7695055543 - 0.2706246407i\)
\(L(1)\) \(\approx\) \(0.5734165585 - 0.3445567827i\)
\(L(1)\) \(\approx\) \(0.5734165585 - 0.3445567827i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.933 + 0.359i)T \)
3 \( 1 + (0.404 - 0.914i)T \)
5 \( 1 + (-0.401 - 0.915i)T \)
7 \( 1 + (-0.197 - 0.980i)T \)
11 \( 1 + (-0.103 - 0.994i)T \)
13 \( 1 + (-0.754 - 0.655i)T \)
17 \( 1 + (0.480 + 0.877i)T \)
19 \( 1 + (0.890 + 0.454i)T \)
23 \( 1 + (-0.477 - 0.878i)T \)
31 \( 1 + (0.0649 - 0.997i)T \)
37 \( 1 + (-0.902 - 0.430i)T \)
41 \( 1 + (0.0909 + 0.995i)T \)
43 \( 1 + (0.680 + 0.733i)T \)
47 \( 1 + (0.530 + 0.847i)T \)
53 \( 1 + (-0.628 + 0.777i)T \)
59 \( 1 + (-0.990 - 0.136i)T \)
61 \( 1 + (0.350 + 0.936i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (0.705 - 0.708i)T \)
73 \( 1 + (0.579 - 0.815i)T \)
79 \( 1 + (0.960 - 0.279i)T \)
83 \( 1 + (0.880 + 0.474i)T \)
89 \( 1 + (0.0714 + 0.997i)T \)
97 \( 1 + (0.563 + 0.826i)T \)
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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

−18.516167633938650655982279726650, −17.73697271102548127756983399481, −17.169765223004546494127798648360, −16.1273560132417018676902396541, −15.60032239470980173344144633683, −15.3922828186068230527880824451, −14.38307983678138201009388818898, −13.89667234246555715164916620887, −12.53040790977301505006647630248, −11.854586602647738871000021899619, −11.53635892957126685168228902445, −10.58936440005306190748284709274, −9.89493296254561316715834587551, −9.52375839116439610511648662627, −8.88617081382478887129899950306, −7.99264661157395312222540730499, −7.26574832113414161433884948070, −6.81636526346109446908870099640, −5.55480097398190160400654402247, −4.84286729059363033713212378489, −3.71558389175300105629517583249, −3.17065151620368813621546722347, −2.403280236808286764780843095060, −1.91809737995490709204556365639, −0.23675105225651124293528235669, 0.70014375795712973833111171975, 0.94573273897223025893392495026, 1.933938951431099280635697124212, 2.98979805247264162033836271316, 3.71811114118947359174408579156, 4.88299147478968787358783172927, 5.87148945381123809025509795504, 6.29307673750466475486771597551, 7.40572650307674063942363786673, 7.94244068004779106906911738454, 8.08364745461632398909524856744, 9.12517869295638688200344575930, 9.68426099144233603783809143435, 10.59375244605274924759327252507, 11.243447817844563462375807570, 12.24394791984221961625287742405, 12.5593640504467182578547047254, 13.57012293997902259026112438844, 14.1079234473994539707943078992, 14.857700867708170300457296080056, 15.662686153692944468278763526670, 16.59150684758475424575694197996, 16.7470883075287932644640317605, 17.54836562497035171309079240821, 18.171030419685498166795481809949

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