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Properties

Label 1-4004-4004.1511-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.991 - 0.126i$
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  + (0.913 − 0.406i)3-s + (−0.309 + 0.951i)5-s + (0.669 − 0.743i)9-s + (0.104 + 0.994i)15-s + (0.978 − 0.207i)17-s + (−0.104 + 0.994i)19-s + (0.5 − 0.866i)23-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (−0.104 − 0.994i)29-s + (0.309 + 0.951i)31-s + (−0.104 − 0.994i)37-s + (−0.913 + 0.406i)41-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)45-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (−0.309 + 0.951i)5-s + (0.669 − 0.743i)9-s + (0.104 + 0.994i)15-s + (0.978 − 0.207i)17-s + (−0.104 + 0.994i)19-s + (0.5 − 0.866i)23-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (−0.104 − 0.994i)29-s + (0.309 + 0.951i)31-s + (−0.104 − 0.994i)37-s + (−0.913 + 0.406i)41-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.518248355 - 0.1599898821i\)
\(L(\frac12)\) \(\approx\) \(2.518248355 - 0.1599898821i\)
\(L(1)\) \(\approx\) \(1.468992318 + 0.03072987807i\)
\(L(1)\) \(\approx\) \(1.468992318 + 0.03072987807i\)

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 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−18.80359837952488205020696842254, −17.721085154625442338581240450724, −16.962875722003272182567456670431, −16.438468973400334697035116181464, −15.62485576856102265049533970597, −15.24497646179963659292693643641, −14.45020637970903450952556653857, −13.66506149006915307878955076794, −13.1026560777302421727506507309, −12.509667950841961358249643650763, −11.60314606511818258127545892091, −10.94644790592944511837311764867, −9.81448825522315139270547945338, −9.60398575055947108891720401176, −8.56871772691698143448698448851, −8.3293385336957403597566109112, −7.431167700713302895802186984897, −6.76549612025150861635509352335, −5.31804947792076919156260142847, −5.13507987644455832512491182177, −4.05857899711201998247125053673, −3.55554053281657667804135918223, −2.66615153900217486262500878169, −1.710110832188844070446887378762, −0.882202569837328411358595038481, 0.793740619344296920176520907113, 1.873058127276054457604862670851, 2.59776308409920326805363939877, 3.3661897825726870882020898362, 3.84962491437241893431645292798, 4.87867887965662396887577014752, 6.02324062105328598888876768004, 6.617818518303771056046006642749, 7.381123823857021002941849321296, 7.94912084348683847972970278511, 8.54117379974001686228922359206, 9.52105429477547305484818732120, 10.13123251536068947641702511862, 10.752568220977766884893262696806, 11.788545809468272503282400929317, 12.26227048086721024753198105015, 13.08057391934034648802341350059, 13.85080380614265559847555730897, 14.56702536972842766368096882295, 14.71665151607649841319193969783, 15.64297305267394654670321969550, 16.28853967816044292261738441883, 17.19499067296055453136422582557, 18.10618845039275214301588036688, 18.555018463012529403328970823712

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