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Properties

Label 1-4024-4024.1139-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.963 + 0.268i$
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  + (−0.704 + 0.709i)3-s + (0.620 + 0.784i)5-s + (0.507 − 0.861i)7-s + (−0.00625 − 0.999i)9-s + (0.988 + 0.149i)11-s + (−0.217 + 0.976i)13-s + (−0.993 − 0.112i)15-s + (−0.560 + 0.828i)17-s + (−0.998 + 0.0500i)19-s + (0.253 + 0.967i)21-s + (0.722 − 0.691i)23-s + (−0.229 + 0.973i)25-s + (0.713 + 0.700i)27-s + (−0.131 + 0.991i)29-s + (−0.943 + 0.331i)31-s + ⋯
L(s)  = 1  + (−0.704 + 0.709i)3-s + (0.620 + 0.784i)5-s + (0.507 − 0.861i)7-s + (−0.00625 − 0.999i)9-s + (0.988 + 0.149i)11-s + (−0.217 + 0.976i)13-s + (−0.993 − 0.112i)15-s + (−0.560 + 0.828i)17-s + (−0.998 + 0.0500i)19-s + (0.253 + 0.967i)21-s + (0.722 − 0.691i)23-s + (−0.229 + 0.973i)25-s + (0.713 + 0.700i)27-s + (−0.131 + 0.991i)29-s + (−0.943 + 0.331i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.963 + 0.268i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.963 + 0.268i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1432341682 + 1.046015792i\)
\(L(\frac12)\) \(\approx\) \(0.1432341682 + 1.046015792i\)
\(L(1)\) \(\approx\) \(0.8334762458 + 0.3995613243i\)
\(L(1)\) \(\approx\) \(0.8334762458 + 0.3995613243i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.704 + 0.709i)T \)
5 \( 1 + (0.620 + 0.784i)T \)
7 \( 1 + (0.507 - 0.861i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (-0.217 + 0.976i)T \)
17 \( 1 + (-0.560 + 0.828i)T \)
19 \( 1 + (-0.998 + 0.0500i)T \)
23 \( 1 + (0.722 - 0.691i)T \)
29 \( 1 + (-0.131 + 0.991i)T \)
31 \( 1 + (-0.943 + 0.331i)T \)
37 \( 1 + (0.241 + 0.970i)T \)
41 \( 1 + (0.277 + 0.960i)T \)
43 \( 1 + (-0.977 + 0.211i)T \)
47 \( 1 + (0.205 - 0.978i)T \)
53 \( 1 + (0.998 + 0.0500i)T \)
59 \( 1 + (-0.982 - 0.186i)T \)
61 \( 1 + (0.803 + 0.595i)T \)
67 \( 1 + (-0.993 + 0.112i)T \)
71 \( 1 + (0.997 + 0.0750i)T \)
73 \( 1 + (-0.871 - 0.490i)T \)
79 \( 1 + (-0.889 + 0.457i)T \)
83 \( 1 + (0.289 - 0.957i)T \)
89 \( 1 + (-0.810 - 0.585i)T \)
97 \( 1 + (-0.883 - 0.468i)T \)
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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

−17.96225053806583331562683135829, −17.48177774463499938732202597854, −17.05730400282072719119064029793, −16.30353916716153236558964749541, −15.47780493831549008838306988676, −14.769492834223449646777312984410, −13.88500970914769001076519751822, −13.25875306020544006800225139758, −12.60498376786616949910780962663, −12.12127702015152101212317173899, −11.34051505077851896445709545963, −10.83904358434188564077601605686, −9.73155955885503024671360450873, −9.06221157594693837995223848566, −8.47155110472336695998985200601, −7.648466179365911483154426995383, −6.84937570866038403273385607636, −5.949045637352949468187165342039, −5.573191304461260161263989374858, −4.88763664001035321088074525158, −4.07614569055629904408879934503, −2.65664490940533618071444637644, −2.04043906012167437830726455639, −1.27967281171755199698538081334, −0.32464966472631907277210408352, 1.293031158453207918615481828, 1.8946302689568469186344268496, 3.1337553134561328685698291796, 3.997716347915723190785967345680, 4.45336063709935697194446956332, 5.26890174476935497978908848505, 6.379210059712488779036217372661, 6.62178560663174958527500195997, 7.26960488176942390810566610911, 8.64883788872051317739424866958, 9.09776781585056689918529211549, 10.109788193010510531587107667066, 10.43801828962734199092526847988, 11.218015221649905755204730235891, 11.56384723430610915264833540384, 12.618409013409555233863032119321, 13.35161940911182675506826010325, 14.2848195025515561194082390347, 14.79294304744278880582750820703, 15.05405638729791263373491463193, 16.397057524234388410176623335799, 16.92875217774267087645802267016, 17.138767479607326375052381885347, 18.024640780238203670470576640748, 18.5510044349187188252157973365

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