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Properties

Label 1-4024-4024.115-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.481 - 0.876i$
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  + (0.974 − 0.223i)3-s + (−0.463 + 0.886i)5-s + (−0.824 − 0.565i)7-s + (0.900 − 0.435i)9-s + (−0.180 − 0.983i)11-s + (0.998 + 0.0625i)13-s + (−0.253 + 0.967i)15-s + (−0.384 + 0.923i)17-s + (0.894 − 0.446i)19-s + (−0.930 − 0.366i)21-s + (0.00625 − 0.999i)23-s + (−0.570 − 0.821i)25-s + (0.780 − 0.625i)27-s + (−0.999 − 0.0375i)29-s + (0.695 + 0.718i)31-s + ⋯
L(s)  = 1  + (0.974 − 0.223i)3-s + (−0.463 + 0.886i)5-s + (−0.824 − 0.565i)7-s + (0.900 − 0.435i)9-s + (−0.180 − 0.983i)11-s + (0.998 + 0.0625i)13-s + (−0.253 + 0.967i)15-s + (−0.384 + 0.923i)17-s + (0.894 − 0.446i)19-s + (−0.930 − 0.366i)21-s + (0.00625 − 0.999i)23-s + (−0.570 − 0.821i)25-s + (0.780 − 0.625i)27-s + (−0.999 − 0.0375i)29-s + (0.695 + 0.718i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.481 - 0.876i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.724011473 - 1.019986937i\)
\(L(\frac12)\) \(\approx\) \(1.724011473 - 1.019986937i\)
\(L(1)\) \(\approx\) \(1.285770701 - 0.1770736743i\)
\(L(1)\) \(\approx\) \(1.285770701 - 0.1770736743i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.974 - 0.223i)T \)
5 \( 1 + (-0.463 + 0.886i)T \)
7 \( 1 + (-0.824 - 0.565i)T \)
11 \( 1 + (-0.180 - 0.983i)T \)
13 \( 1 + (0.998 + 0.0625i)T \)
17 \( 1 + (-0.384 + 0.923i)T \)
19 \( 1 + (0.894 - 0.446i)T \)
23 \( 1 + (0.00625 - 0.999i)T \)
29 \( 1 + (-0.999 - 0.0375i)T \)
31 \( 1 + (0.695 + 0.718i)T \)
37 \( 1 + (0.289 + 0.957i)T \)
41 \( 1 + (-0.143 + 0.989i)T \)
43 \( 1 + (-0.925 - 0.378i)T \)
47 \( 1 + (0.668 - 0.743i)T \)
53 \( 1 + (-0.894 - 0.446i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (0.395 - 0.918i)T \)
67 \( 1 + (-0.253 - 0.967i)T \)
71 \( 1 + (0.640 - 0.768i)T \)
73 \( 1 + (0.731 - 0.682i)T \)
79 \( 1 + (0.951 + 0.307i)T \)
83 \( 1 + (-0.686 + 0.726i)T \)
89 \( 1 + (-0.474 - 0.880i)T \)
97 \( 1 + (-0.831 - 0.554i)T \)
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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

−18.700830733004860645976603980262, −18.09773975449226894484249277203, −17.15984715121798606373649735252, −16.14669427768880893209489568556, −15.83645719176574432584629506116, −15.433133974484580971676118424672, −14.594922647406490516958495302694, −13.59761809213941254181910071737, −13.23320904016374829675418744274, −12.555350042876152499972264909756, −11.86410295798453924875228523566, −11.08171553585459512637875154211, −9.91241544327714188665999843780, −9.44814782172141228554171919224, −9.03989479692909509969046229117, −8.16407501768440711235827382770, −7.552720268503939080589879728385, −6.88054997740400526625801096803, −5.69035529390208448260702798058, −5.11261495366189694844767060039, −4.090665285886945085791308784297, −3.658186343258335247770380904261, −2.75240152450915670952837590134, −1.935718476964350209609470392894, −1.00896679887697360233205128019, 0.54834649849787327412769459565, 1.59423344885985357067873349791, 2.70508935234715016013042960329, 3.34008075984842864277789218302, 3.664798652536039790243861387393, 4.57284418219839657762317010254, 5.9597240328942854363939039428, 6.61676686973516193914099456065, 7.01406378285502191028052605256, 8.145886588627624200695604436707, 8.322747923741460414008150214351, 9.33053413202699564755808923623, 10.07945413220261364897510824213, 10.74767800972920530774890836114, 11.32221803367374247442270916545, 12.32158457670759806256279579884, 13.136160886889905394883618787108, 13.68968286915873658086991758088, 14.05869927701659461105950212620, 15.09654545829204060195521774406, 15.44709735883043309260170389857, 16.23452552484354590865373714697, 16.80412699263709583920639666450, 18.06210576108142910357031414303, 18.46777534517087815751966416899

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