↑

Properties

Label 1-4024-4024.141-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.221 - 0.975i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.889 + 0.457i)3-s + (−0.313 + 0.949i)5-s + (0.677 − 0.735i)7-s + (0.580 − 0.814i)9-s + (0.668 + 0.743i)11-s + (0.301 − 0.953i)13-s + (−0.155 − 0.987i)15-s + (−0.739 + 0.673i)17-s + (−0.241 + 0.970i)19-s + (−0.265 + 0.964i)21-s + (−0.982 − 0.186i)23-s + (−0.803 − 0.595i)25-s + (−0.143 + 0.989i)27-s + (−0.429 − 0.902i)29-s + (0.452 − 0.891i)31-s + ⋯
L(s)  = 1  + (−0.889 + 0.457i)3-s + (−0.313 + 0.949i)5-s + (0.677 − 0.735i)7-s + (0.580 − 0.814i)9-s + (0.668 + 0.743i)11-s + (0.301 − 0.953i)13-s + (−0.155 − 0.987i)15-s + (−0.739 + 0.673i)17-s + (−0.241 + 0.970i)19-s + (−0.265 + 0.964i)21-s + (−0.982 − 0.186i)23-s + (−0.803 − 0.595i)25-s + (−0.143 + 0.989i)27-s + (−0.429 − 0.902i)29-s + (0.452 − 0.891i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.221 - 0.975i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.221 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4770013286 - 0.3809483876i\)
\(L(\frac12)\) \(\approx\) \(0.4770013286 - 0.3809483876i\)
\(L(1)\) \(\approx\) \(0.7355159213 + 0.1045139812i\)
\(L(1)\) \(\approx\) \(0.7355159213 + 0.1045139812i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.889 + 0.457i)T \)
5 \( 1 + (-0.313 + 0.949i)T \)
7 \( 1 + (0.677 - 0.735i)T \)
11 \( 1 + (0.668 + 0.743i)T \)
13 \( 1 + (0.301 - 0.953i)T \)
17 \( 1 + (-0.739 + 0.673i)T \)
19 \( 1 + (-0.241 + 0.970i)T \)
23 \( 1 + (-0.982 - 0.186i)T \)
29 \( 1 + (-0.429 - 0.902i)T \)
31 \( 1 + (0.452 - 0.891i)T \)
37 \( 1 + (-0.824 - 0.565i)T \)
41 \( 1 + (0.384 + 0.923i)T \)
43 \( 1 + (-0.600 + 0.799i)T \)
47 \( 1 + (0.999 - 0.0250i)T \)
53 \( 1 + (-0.241 - 0.970i)T \)
59 \( 1 + (0.965 + 0.259i)T \)
61 \( 1 + (0.934 - 0.355i)T \)
67 \( 1 + (0.155 - 0.987i)T \)
71 \( 1 + (0.407 - 0.913i)T \)
73 \( 1 + (-0.858 + 0.512i)T \)
79 \( 1 + (-0.999 + 0.0375i)T \)
83 \( 1 + (-0.747 - 0.663i)T \)
89 \( 1 + (0.640 + 0.768i)T \)
97 \( 1 + (0.360 - 0.932i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.73993384599688135370531662104, −17.69636424689149514096536920205, −17.40931682392150155766862050021, −16.60738819312193473661137065774, −15.854672175217329298821897897303, −15.672981366035809449758887436335, −14.35872104815460074852063159723, −13.74755448532386277595958278425, −13.11346441219696534976580730533, −12.15196266970331547025734146685, −11.833167862200377624876125850693, −11.32291945008683570851325756939, −10.6180312225563723987628582944, −9.40944854931729082216683582874, −8.71149421363696091276777038085, −8.437490403766745817398464092672, −7.22547654574597656234711090747, −6.7390200573635314446269947462, −5.76424385340517875574956659963, −5.25056285292846040773811209150, −4.51698413852639042129008841536, −3.88242060523797764359978396458, −2.47432766248422186954859338186, −1.65482348934823120696575813177, −0.99177279453063918029918827669, 0.22178438354652132673057297597, 1.439151769380448689154383215491, 2.27032365535596293096005595092, 3.66042583046314497443480933623, 3.98707260046839214350061359606, 4.65251328665751284074006108426, 5.775300073668633815869538255725, 6.28340236027818330376996983995, 7.016046064298375419870028791720, 7.78990872710875886162789828269, 8.391114357171768577468229094928, 9.77438437742341428582524462265, 10.09619493508646525009267995097, 10.799577857126762640339620658780, 11.33746729331096186977907696794, 11.94566359665432802409685042446, 12.732773082972349883688243079312, 13.56876607367561457355972474501, 14.56829328002910025688335588442, 14.88084770527163382902994380110, 15.56004547048030235461914815708, 16.30002897347589058626142795073, 17.19818390166251329326161948537, 17.57713248488221706222464335073, 18.085501084760027794459056540924

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