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Properties

Label 1-4024-4024.1059-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.700 + 0.713i$
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  + (−0.590 + 0.806i)3-s + (0.984 + 0.174i)5-s + (0.803 − 0.595i)7-s + (−0.301 − 0.953i)9-s + (0.474 + 0.880i)11-s + (0.965 − 0.259i)13-s + (−0.722 + 0.691i)15-s + (0.756 − 0.654i)17-s + (0.772 + 0.635i)19-s + (0.00625 + 0.999i)21-s + (0.958 + 0.283i)23-s + (0.939 + 0.343i)25-s + (0.947 + 0.319i)27-s + (−0.155 + 0.987i)29-s + (0.659 − 0.752i)31-s + ⋯
L(s)  = 1  + (−0.590 + 0.806i)3-s + (0.984 + 0.174i)5-s + (0.803 − 0.595i)7-s + (−0.301 − 0.953i)9-s + (0.474 + 0.880i)11-s + (0.965 − 0.259i)13-s + (−0.722 + 0.691i)15-s + (0.756 − 0.654i)17-s + (0.772 + 0.635i)19-s + (0.00625 + 0.999i)21-s + (0.958 + 0.283i)23-s + (0.939 + 0.343i)25-s + (0.947 + 0.319i)27-s + (−0.155 + 0.987i)29-s + (0.659 − 0.752i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.700 + 0.713i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.700 + 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.323513271 + 0.9751761318i\)
\(L(\frac12)\) \(\approx\) \(2.323513271 + 0.9751761318i\)
\(L(1)\) \(\approx\) \(1.342672804 + 0.3376438898i\)
\(L(1)\) \(\approx\) \(1.342672804 + 0.3376438898i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.590 + 0.806i)T \)
5 \( 1 + (0.984 + 0.174i)T \)
7 \( 1 + (0.803 - 0.595i)T \)
11 \( 1 + (0.474 + 0.880i)T \)
13 \( 1 + (0.965 - 0.259i)T \)
17 \( 1 + (0.756 - 0.654i)T \)
19 \( 1 + (0.772 + 0.635i)T \)
23 \( 1 + (0.958 + 0.283i)T \)
29 \( 1 + (-0.155 + 0.987i)T \)
31 \( 1 + (0.659 - 0.752i)T \)
37 \( 1 + (-0.570 + 0.821i)T \)
41 \( 1 + (0.943 + 0.331i)T \)
43 \( 1 + (0.539 - 0.842i)T \)
47 \( 1 + (-0.640 + 0.768i)T \)
53 \( 1 + (-0.772 + 0.635i)T \)
59 \( 1 + (0.974 - 0.223i)T \)
61 \( 1 + (0.990 - 0.137i)T \)
67 \( 1 + (-0.722 - 0.691i)T \)
71 \( 1 + (-0.858 - 0.512i)T \)
73 \( 1 + (-0.999 - 0.0125i)T \)
79 \( 1 + (0.253 - 0.967i)T \)
83 \( 1 + (0.962 + 0.271i)T \)
89 \( 1 + (-0.731 + 0.682i)T \)
97 \( 1 + (-0.349 + 0.937i)T \)
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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

−18.25657587150871095928403776696, −17.60454815670033321184760440513, −17.36595178451945113005913018158, −16.38078337582672233760990744379, −15.976716573289170428027343884285, −14.70202078176843273417704704384, −14.21902347951520894628969822579, −13.522396626790879887866002946867, −13.010559709563330570035005701754, −12.19669581087895969637552850082, −11.45965289634500401644170406553, −11.04124725115018325461936466310, −10.23873998376668050653784671001, −9.189451863520732236520977443333, −8.60844440223497427625497680459, −8.0293652333551840089592096009, −6.99125403053289166967634072838, −6.312742564772204396579704083146, −5.63532711064100519691982501489, −5.30583833132957851744640610723, −4.26877044105463936089152435297, −3.055345625111016071448754557296, −2.29058924056832949598633780528, −1.31077198872539055198340108907, −1.02704244299193759341981765566, 1.113838338356041529901831537060, 1.44538158515820009537715873796, 2.83054566947146602706744081548, 3.58009623818049191498827109458, 4.44983803998265864199616501902, 5.15235147642592425580919435050, 5.664907823145024318688154871125, 6.53674702810524555607730740410, 7.22549369732425287829584137960, 8.12008661738060651775240826897, 9.194461506690134324383678593506, 9.56684511458364094825474300071, 10.37396446975615421894300352728, 10.82110100738721992001714472086, 11.5875250317971062445663896825, 12.23383235353892991157240304693, 13.18766938133399694386993657344, 13.92268898245703992236426944903, 14.55077445169029826155901224119, 15.01812694785645566697459188773, 16.0127278394737909995267450449, 16.57024009220410425067388865148, 17.31660547149263643964876190064, 17.70393371856084192971992466150, 18.25555667691759325755115538422

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