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Properties

Label 1-4004-4004.2243-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.362 + 0.931i$
Analytic cond. $18.5944$
Root an. cond. $18.5944$
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  + (−0.5 + 0.866i)3-s + (0.866 + 0.5i)5-s + (−0.5 − 0.866i)9-s + (−0.866 + 0.5i)15-s − 17-s + (−0.866 + 0.5i)19-s + 23-s + (0.5 + 0.866i)25-s + 27-s + (0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s − i·37-s + (0.866 − 0.5i)41-s + (0.5 − 0.866i)43-s − i·45-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.866 + 0.5i)5-s + (−0.5 − 0.866i)9-s + (−0.866 + 0.5i)15-s − 17-s + (−0.866 + 0.5i)19-s + 23-s + (0.5 + 0.866i)25-s + 27-s + (0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s − i·37-s + (0.866 − 0.5i)41-s + (0.5 − 0.866i)43-s − i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.362 + 0.931i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (2243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.362 + 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.352716815 + 0.9251832715i\)
\(L(\frac12)\) \(\approx\) \(1.352716815 + 0.9251832715i\)
\(L(1)\) \(\approx\) \(0.9903917112 + 0.3613967989i\)
\(L(1)\) \(\approx\) \(0.9903917112 + 0.3613967989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 - iT \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + iT \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−18.24687264462162777838685187367, −17.46537433396373165943114289228, −17.29235014105567483569820771767, −16.55731454417066506007737126230, −15.709428008998383162675172395752, −14.93257054048021571505682773417, −13.95293338822259913666644178573, −13.46137817165254255253136153968, −12.95200481353512436019869213976, −12.3170936948089412644202061749, −11.53024940203861505330102419691, −10.811182379135713072917751256658, −10.18999737016097601848826855652, −9.1540784030094041594518057472, −8.66558393420505965350556330684, −7.8637556084785111154134259459, −6.93589590672202681726014675819, −6.32678662918551239107344768145, −5.85514691003080340147884915517, −4.76323195394150800885764662731, −4.504117497653394821521106388234, −2.83571649403995739210005033225, −2.35822989286047655559424544468, −1.41992635378039250649082775834, −0.68829238591249292687803350547, 0.76607538680573210381623302708, 2.01763570216952324556467734544, 2.72483629002135807850364033734, 3.64542386887082968319187612145, 4.42052416746686014920538320870, 5.17660344422194114404616066831, 5.894555249666893350405975794, 6.52628883274722510176374401855, 7.14786231589056728357563331867, 8.40757864746899878972932157477, 9.082116047066832461362365495986, 9.63281289206898975341641070769, 10.510820767580695986178232900996, 10.8282761934571207057920286087, 11.48721081913106729393410397367, 12.56191444032850638169051563428, 13.012316047574037104153509469686, 14.191960818235446721729035657793, 14.37619122771528006179306943436, 15.38386949848053819521107929415, 15.7568431788527009088960030850, 16.701459876465571766313430721876, 17.410569374233468298120377540214, 17.56645278791402698679899809838, 18.55832126118981624660961015056

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