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Properties

Label 1-4029-4029.1256-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.443 - 0.896i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (−0.663 + 0.748i)2-s + (−0.120 − 0.992i)4-s + (0.855 − 0.517i)5-s + (0.180 − 0.983i)7-s + (0.822 + 0.568i)8-s + (−0.180 + 0.983i)10-s + (−0.855 − 0.517i)11-s + (−0.120 − 0.992i)13-s + (0.616 + 0.787i)14-s + (−0.970 + 0.239i)16-s + (0.464 + 0.885i)19-s + (−0.616 − 0.787i)20-s + (0.954 − 0.297i)22-s + (0.707 + 0.707i)23-s + (0.464 − 0.885i)25-s + (0.822 + 0.568i)26-s + ⋯
L(s)  = 1  + (−0.663 + 0.748i)2-s + (−0.120 − 0.992i)4-s + (0.855 − 0.517i)5-s + (0.180 − 0.983i)7-s + (0.822 + 0.568i)8-s + (−0.180 + 0.983i)10-s + (−0.855 − 0.517i)11-s + (−0.120 − 0.992i)13-s + (0.616 + 0.787i)14-s + (−0.970 + 0.239i)16-s + (0.464 + 0.885i)19-s + (−0.616 − 0.787i)20-s + (0.954 − 0.297i)22-s + (0.707 + 0.707i)23-s + (0.464 − 0.885i)25-s + (0.822 + 0.568i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.443 - 0.896i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (1256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.443 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5332270921 - 0.8591664426i\)
\(L(\frac12)\) \(\approx\) \(0.5332270921 - 0.8591664426i\)
\(L(1)\) \(\approx\) \(0.8173663646 - 0.1100049578i\)
\(L(1)\) \(\approx\) \(0.8173663646 - 0.1100049578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.663 + 0.748i)T \)
5 \( 1 + (0.855 - 0.517i)T \)
7 \( 1 + (0.180 - 0.983i)T \)
11 \( 1 + (-0.855 - 0.517i)T \)
13 \( 1 + (-0.120 - 0.992i)T \)
19 \( 1 + (0.464 + 0.885i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.911 - 0.410i)T \)
31 \( 1 + (0.998 - 0.0603i)T \)
37 \( 1 + (0.297 - 0.954i)T \)
41 \( 1 + (-0.517 - 0.855i)T \)
43 \( 1 + (0.239 - 0.970i)T \)
47 \( 1 + (-0.885 - 0.464i)T \)
53 \( 1 + (0.822 + 0.568i)T \)
59 \( 1 + (-0.992 - 0.120i)T \)
61 \( 1 + (-0.954 - 0.297i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (0.180 + 0.983i)T \)
73 \( 1 + (-0.616 - 0.787i)T \)
83 \( 1 + (0.992 - 0.120i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (0.297 - 0.954i)T \)
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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

−18.636388375531651202410440014262, −18.10719103834224562249156729831, −17.64581726086863662112718187622, −16.80289155771574302021799733908, −16.185759988157838854229206277666, −15.20650485394783177491531903369, −14.71634712179099988750812524312, −13.63019219822962089590151474838, −13.20515299357255822587123821995, −12.4272692727205033876834896077, −11.67721573007971746760744271890, −11.096265389175934061485418528046, −10.40078946424654222016582577878, −9.55626819830697047815681927548, −9.27793226001223467748364203517, −8.44645063737870824149988270030, −7.62585298901180869767554489418, −6.818200037749457410789777648514, −6.18240300828435536697001299533, −4.965355801139419569810937700494, −4.64149239910965258481138254139, −3.120969336411936365583380207553, −2.73946914513260842426268524723, −2.012303583092741703659718173264, −1.307592083388145007867197788501, 0.36861695917055838794727521057, 1.15145622166865480263781141706, 1.97356818625689556714270900212, 3.06438521010414209534832722811, 4.14110591115286558521963460241, 5.16509790142560884421512450627, 5.51851709425517741924494749904, 6.21263504436975379443588968173, 7.25846544701034190463569064587, 7.75043155771527713089745304186, 8.39740032016611647497617351266, 9.21817909974119247795592720396, 9.94725216848328318998780988549, 10.46332120688025815638497789488, 10.96279964479780498418658958015, 12.109930659414573834756243340429, 13.2147596087695028873442567083, 13.47720206691623287713934173122, 14.13649196030102158680997430667, 14.96773245239076106293805037332, 15.66943260564611604364684030307, 16.39888654721576911870070688408, 16.95142901463146675702209791487, 17.4718935154126963220338801693, 18.05008336153538755572971869287

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