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Properties

Label 1-4004-4004.1371-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.211 + 0.977i$
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  + (−0.104 + 0.994i)3-s + (0.951 + 0.309i)5-s + (−0.978 − 0.207i)9-s + (−0.406 + 0.913i)15-s + (−0.669 − 0.743i)17-s + (0.406 + 0.913i)19-s + (−0.5 − 0.866i)23-s + (0.809 + 0.587i)25-s + (0.309 − 0.951i)27-s + (−0.913 − 0.406i)29-s + (0.951 − 0.309i)31-s + (−0.406 + 0.913i)37-s + (0.994 + 0.104i)41-s + (0.5 − 0.866i)43-s + (−0.866 − 0.5i)45-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)3-s + (0.951 + 0.309i)5-s + (−0.978 − 0.207i)9-s + (−0.406 + 0.913i)15-s + (−0.669 − 0.743i)17-s + (0.406 + 0.913i)19-s + (−0.5 − 0.866i)23-s + (0.809 + 0.587i)25-s + (0.309 − 0.951i)27-s + (−0.913 − 0.406i)29-s + (0.951 − 0.309i)31-s + (−0.406 + 0.913i)37-s + (0.994 + 0.104i)41-s + (0.5 − 0.866i)43-s + (−0.866 − 0.5i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.470824610 + 1.186882490i\)
\(L(\frac12)\) \(\approx\) \(1.470824610 + 1.186882490i\)
\(L(1)\) \(\approx\) \(1.103498242 + 0.4320277253i\)
\(L(1)\) \(\approx\) \(1.103498242 + 0.4320277253i\)

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

−18.05077069857452159037870068505, −17.640452519467356275683453736367, −17.370736759386792212066027654871, −16.38328956186298925386794475825, −15.73584119917847507092943424041, −14.68746947539862798601650919283, −14.114196087744491951349044094256, −13.41322370400698275793716181091, −12.9615266271194973601822427131, −12.37951589726382376992592931773, −11.433201017353291868476352524769, −10.92535168784570326836814541444, −9.976006042241650414166747016866, −9.17394025590335050820188474979, −8.658616958021533046288924851035, −7.77383903863362555416332396039, −7.05596607535722359688446511882, −6.33770440848481452764187259709, −5.71445133442538220389833108518, −5.09136742421860832202421573584, −4.066062399979500534892306582612, −2.94655625996508339190507193397, −2.199630030625828482808661959215, −1.56842612821750544729875179053, −0.66911572383579972490855237667, 0.8373678686448446438923406989, 2.17705496772404349422246641629, 2.66639146464803656686673399000, 3.67241479809911849577018654256, 4.375413527007355845003778804863, 5.23131965036581896891005564558, 5.82883184986463532977900290244, 6.478807942320281851236911103173, 7.38507267293766573140394614350, 8.4161017272700942826262289309, 9.04366655675018188788865383946, 9.808668219760461444536857236428, 10.1876289205242444586435675147, 10.91501063562341366879856103987, 11.64159165310813907976052474320, 12.358164768124961391980488115968, 13.42243048792669743159016206621, 13.92284504985372310596597428138, 14.54872989263171734745177268023, 15.24312661649202041037566389140, 15.92022929935768660505776305882, 16.65785735552813599232951872977, 17.14773609344190492357435290450, 17.89661304248690371760698455629, 18.48802011811055433199183980455

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