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Properties

Label 1-3997-3997.796-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.627 - 0.778i$
Analytic cond. $429.537$
Root an. cond. $429.537$
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  + (−0.926 − 0.376i)2-s + (−0.411 + 0.911i)3-s + (0.716 + 0.697i)4-s + (−0.999 − 0.0220i)5-s + (0.724 − 0.689i)6-s + (−0.401 − 0.915i)8-s + (−0.660 − 0.750i)9-s + (0.917 + 0.396i)10-s + (0.802 + 0.596i)11-s + (−0.930 + 0.366i)12-s + (0.934 − 0.355i)13-s + (0.431 − 0.901i)15-s + (0.0275 + 0.999i)16-s + (0.381 − 0.924i)17-s + (0.329 + 0.944i)18-s + (−0.952 + 0.303i)19-s + ⋯
L(s)  = 1  + (−0.926 − 0.376i)2-s + (−0.411 + 0.911i)3-s + (0.716 + 0.697i)4-s + (−0.999 − 0.0220i)5-s + (0.724 − 0.689i)6-s + (−0.401 − 0.915i)8-s + (−0.660 − 0.750i)9-s + (0.917 + 0.396i)10-s + (0.802 + 0.596i)11-s + (−0.930 + 0.366i)12-s + (0.934 − 0.355i)13-s + (0.431 − 0.901i)15-s + (0.0275 + 0.999i)16-s + (0.381 − 0.924i)17-s + (0.329 + 0.944i)18-s + (−0.952 + 0.303i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.627 - 0.778i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (796, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.627 - 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2641485284 - 0.5517874035i\)
\(L(\frac12)\) \(\approx\) \(0.2641485284 - 0.5517874035i\)
\(L(1)\) \(\approx\) \(0.5452662198 + 0.02343938828i\)
\(L(1)\) \(\approx\) \(0.5452662198 + 0.02343938828i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.926 - 0.376i)T \)
3 \( 1 + (-0.411 + 0.911i)T \)
5 \( 1 + (-0.999 - 0.0220i)T \)
11 \( 1 + (0.802 + 0.596i)T \)
13 \( 1 + (0.934 - 0.355i)T \)
17 \( 1 + (0.381 - 0.924i)T \)
19 \( 1 + (-0.952 + 0.303i)T \)
23 \( 1 + (0.202 - 0.979i)T \)
29 \( 1 + (0.546 - 0.837i)T \)
31 \( 1 + (-0.851 + 0.523i)T \)
37 \( 1 + (-0.889 + 0.456i)T \)
41 \( 1 + (0.0825 + 0.996i)T \)
43 \( 1 + (0.490 - 0.871i)T \)
47 \( 1 + (-0.0275 - 0.999i)T \)
53 \( 1 + (0.528 - 0.849i)T \)
59 \( 1 + (0.754 - 0.656i)T \)
61 \( 1 + (0.644 - 0.764i)T \)
67 \( 1 + (0.528 - 0.849i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.840 - 0.542i)T \)
79 \( 1 + (-0.441 - 0.897i)T \)
83 \( 1 + (0.724 - 0.689i)T \)
89 \( 1 + (-0.959 - 0.282i)T \)
97 \( 1 + (0.768 - 0.639i)T \)
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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

−18.77047731492312743319244911562, −17.76812698271086297473076194191, −17.33756997855582906962023748116, −16.508339778762171683151230212760, −16.16151232908404009660235420168, −15.296898405469169437464559433129, −14.54431116337205719948639249653, −13.94796107891467524329483368912, −12.9140055756292040601177559103, −12.25746131155452693481236899726, −11.4646963553165522658220793425, −11.05360733322132647767040618647, −10.53643247852010257827189815290, −9.23763068223997990000325381880, −8.50491237409096118081116052352, −8.28972131969004142462129938278, −7.16273677401189327236379778949, −6.96405253140325583686387212351, −5.979340398790852404054486526133, −5.57307950234013622278665203366, −4.244215735948412666295221980803, −3.446278617789386709880237485712, −2.36489881124791162084526982066, −1.30546419363576529508236266060, −0.95782583486914133293334190128, 0.2172187226198128605478574134, 0.74355865785229575318961006752, 1.92571845765792793570007053282, 3.07909138174292924775588256608, 3.6696785540172638544288326735, 4.28717088243612677913533092728, 5.13774500896073366907765016579, 6.366036581993363158813917443383, 6.79480993087710854258937880552, 7.77536919500005051897405585239, 8.62394658712677304928690055646, 8.89897011398752998969507237628, 9.91654785983570301288695004262, 10.42961117297715497002445211857, 11.124444726566578474677288416389, 11.71465287898494012648012153089, 12.21389140344983023741914727586, 12.93264870916178246385400583468, 14.24178209507111099579556827729, 15.002326506363257149092944827000, 15.520675243226089719186732159666, 16.23421507612238764664707683305, 16.62731278395871517526045499968, 17.351013672929862771636627872179, 18.06693694082696915957187770378

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