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Properties

Label 1-4024-4024.1149-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.999 + 0.0243i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (0.858 + 0.512i)3-s + (0.485 − 0.874i)5-s + (−0.925 + 0.378i)7-s + (0.474 + 0.880i)9-s + (−0.764 + 0.644i)11-s + (−0.999 − 0.0250i)13-s + (0.864 − 0.501i)15-s + (−0.155 − 0.987i)17-s + (0.686 − 0.726i)19-s + (−0.988 − 0.149i)21-s + (0.810 − 0.585i)23-s + (−0.528 − 0.848i)25-s + (−0.0437 + 0.999i)27-s + (0.817 − 0.575i)29-s + (−0.00625 + 0.999i)31-s + ⋯
L(s)  = 1  + (0.858 + 0.512i)3-s + (0.485 − 0.874i)5-s + (−0.925 + 0.378i)7-s + (0.474 + 0.880i)9-s + (−0.764 + 0.644i)11-s + (−0.999 − 0.0250i)13-s + (0.864 − 0.501i)15-s + (−0.155 − 0.987i)17-s + (0.686 − 0.726i)19-s + (−0.988 − 0.149i)21-s + (0.810 − 0.585i)23-s + (−0.528 − 0.848i)25-s + (−0.0437 + 0.999i)27-s + (0.817 − 0.575i)29-s + (−0.00625 + 0.999i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.999 + 0.0243i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.999 + 0.0243i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.101248479 + 0.02563331322i\)
\(L(\frac12)\) \(\approx\) \(2.101248479 + 0.02563331322i\)
\(L(1)\) \(\approx\) \(1.318725259 + 0.07823475876i\)
\(L(1)\) \(\approx\) \(1.318725259 + 0.07823475876i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.858 + 0.512i)T \)
5 \( 1 + (0.485 - 0.874i)T \)
7 \( 1 + (-0.925 + 0.378i)T \)
11 \( 1 + (-0.764 + 0.644i)T \)
13 \( 1 + (-0.999 - 0.0250i)T \)
17 \( 1 + (-0.155 - 0.987i)T \)
19 \( 1 + (0.686 - 0.726i)T \)
23 \( 1 + (0.810 - 0.585i)T \)
29 \( 1 + (0.817 - 0.575i)T \)
31 \( 1 + (-0.00625 + 0.999i)T \)
37 \( 1 + (0.418 + 0.908i)T \)
41 \( 1 + (-0.253 - 0.967i)T \)
43 \( 1 + (-0.452 + 0.891i)T \)
47 \( 1 + (-0.570 + 0.821i)T \)
53 \( 1 + (0.686 + 0.726i)T \)
59 \( 1 + (-0.640 + 0.768i)T \)
61 \( 1 + (0.986 + 0.161i)T \)
67 \( 1 + (-0.864 - 0.501i)T \)
71 \( 1 + (0.939 - 0.343i)T \)
73 \( 1 + (0.955 - 0.295i)T \)
79 \( 1 + (0.992 + 0.124i)T \)
83 \( 1 + (-0.947 + 0.319i)T \)
89 \( 1 + (0.677 - 0.735i)T \)
97 \( 1 + (-0.649 - 0.760i)T \)
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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

−18.5775630892858712837274568075, −17.99084895800370233567553451217, −17.168347983964322498589627234104, −16.470830335029300070640419503, −15.558343453762915152863232516192, −14.95697682248419538527986884403, −14.343836647242581044944826330500, −13.65340990357380968437904520272, −13.13165298848292845850253838777, −12.593630803826980248930731496022, −11.639189927021457299678359075362, −10.72333217974221948735671206099, −9.96299798520910281699258130564, −9.68407809823945238778068373058, −8.70518558224150544806489864545, −7.86793824341455331381839499562, −7.284956098111192969942566948250, −6.63448147666839956017234266159, −5.98120241496186537420133091250, −5.11345169175400434268795194086, −3.71827244620113883849032042957, −3.36797559819132148557041726963, −2.57827796845517095501700413598, −1.95118247033558710580671919941, −0.76688619241507436890913120424, 0.67808495100831344110025049373, 1.95244371980069376161045459055, 2.786258145100597257326915987530, 3.01924088169410012293188835750, 4.51220759435548635098273113772, 4.82086845328329012518836153139, 5.47769722228818247281841738606, 6.67281546108119510695280430184, 7.32027079809543153990553908957, 8.162702316729664685360197262, 8.95146350768174436804228196876, 9.4795291357337990409213169193, 9.911118230644344178147738980379, 10.57808492220723240646334691457, 11.80654175466056344735723494236, 12.49101847043415304508170578208, 13.08026683154575203803479039019, 13.6537410701506348609943462058, 14.31152177085715380981774812067, 15.38476423688684906584234563042, 15.56797109666386177419108666262, 16.394725834742867029109720317771, 16.894966733579672252757577639095, 17.88764423317081825368022008876, 18.437651160377371135785276403830

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