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Properties

Label 1-4024-4024.1251-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.977 - 0.209i$
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  + (0.0687 − 0.997i)3-s + (0.560 − 0.828i)5-s + (0.649 + 0.760i)7-s + (−0.990 − 0.137i)9-s + (−0.986 + 0.161i)11-s + (0.106 + 0.994i)13-s + (−0.787 − 0.615i)15-s + (0.871 − 0.490i)17-s + (−0.452 − 0.891i)19-s + (0.803 − 0.595i)21-s + (0.463 + 0.886i)23-s + (−0.372 − 0.928i)25-s + (−0.205 + 0.978i)27-s + (0.968 + 0.247i)29-s + (0.407 + 0.913i)31-s + ⋯
L(s)  = 1  + (0.0687 − 0.997i)3-s + (0.560 − 0.828i)5-s + (0.649 + 0.760i)7-s + (−0.990 − 0.137i)9-s + (−0.986 + 0.161i)11-s + (0.106 + 0.994i)13-s + (−0.787 − 0.615i)15-s + (0.871 − 0.490i)17-s + (−0.452 − 0.891i)19-s + (0.803 − 0.595i)21-s + (0.463 + 0.886i)23-s + (−0.372 − 0.928i)25-s + (−0.205 + 0.978i)27-s + (0.968 + 0.247i)29-s + (0.407 + 0.913i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.977 - 0.209i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.977 - 0.209i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.892279645 - 0.2006489305i\)
\(L(\frac12)\) \(\approx\) \(1.892279645 - 0.2006489305i\)
\(L(1)\) \(\approx\) \(1.174843048 - 0.3012656713i\)
\(L(1)\) \(\approx\) \(1.174843048 - 0.3012656713i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.0687 - 0.997i)T \)
5 \( 1 + (0.560 - 0.828i)T \)
7 \( 1 + (0.649 + 0.760i)T \)
11 \( 1 + (-0.986 + 0.161i)T \)
13 \( 1 + (0.106 + 0.994i)T \)
17 \( 1 + (0.871 - 0.490i)T \)
19 \( 1 + (-0.452 - 0.891i)T \)
23 \( 1 + (0.463 + 0.886i)T \)
29 \( 1 + (0.968 + 0.247i)T \)
31 \( 1 + (0.407 + 0.913i)T \)
37 \( 1 + (-0.610 + 0.791i)T \)
41 \( 1 + (0.996 - 0.0875i)T \)
43 \( 1 + (-0.0312 - 0.999i)T \)
47 \( 1 + (-0.168 + 0.985i)T \)
53 \( 1 + (0.452 - 0.891i)T \)
59 \( 1 + (-0.549 + 0.835i)T \)
61 \( 1 + (-0.0937 + 0.995i)T \)
67 \( 1 + (-0.787 + 0.615i)T \)
71 \( 1 + (-0.0812 - 0.996i)T \)
73 \( 1 + (0.289 + 0.957i)T \)
79 \( 1 + (0.507 + 0.861i)T \)
83 \( 1 + (-0.982 + 0.186i)T \)
89 \( 1 + (-0.360 + 0.932i)T \)
97 \( 1 + (-0.253 + 0.967i)T \)
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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

−18.38496753793909789819251971580, −17.746989017512081106725571405045, −17.08424770427396709359857609171, −16.56441270390496237672330180610, −15.65319799029137721381263694854, −15.04033993527737353279146269668, −14.452085657535454802052377695, −13.9839468713041859177860719152, −13.15770401325937196883431230208, −12.35815169513154780459170801435, −11.22622149633779322167140744096, −10.735966490154665001340269311183, −10.21965722611295188944258680075, −9.929112112496846876505844901186, −8.71521850569765200828791824890, −7.97448144519838853077352300220, −7.56514027926957930966842721959, −6.28630675960620815143476993745, −5.7667217407335453097030749672, −5.00486380387210435126560879978, −4.23308109989006179774441198076, −3.35988901770055636683088926279, −2.80379515446647977834620257465, −1.89290158699010334389104745115, −0.585271300214689353584347609507, 0.97601179641625315603998401420, 1.59736960053714086939459108553, 2.42606741700763250664026610453, 2.980073520497479162796837488805, 4.45920357815185575065000429822, 5.1787487873161533953667917757, 5.59641970186750366969982139449, 6.53047502403480912929668258343, 7.2713894119166136409656151335, 8.05567281206962358464617785180, 8.72888156425116845129744628422, 9.13315893217436521323101447342, 10.100334197667120704984158316809, 11.06692172996340731302261302971, 11.82429567093518099855959430858, 12.29923016687861682193002667800, 12.935033604618965552682241369461, 13.76871191966168859682507274262, 14.01779035841981932769888454030, 15.01200357801980897933924680441, 15.77285250436018928104347356217, 16.5013874861881564043613101351, 17.319805177236095508808500389322, 17.83264597346015492094474782871, 18.30244144867957399873415830489

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