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Properties

Label 1-6004-6004.1507-r0-0-0
Degree $1$
Conductor $6004$
Sign $0.999 + 0.0426i$
Analytic cond. $27.8824$
Root an. cond. $27.8824$
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  + (−0.452 − 0.891i)3-s + (−0.872 + 0.488i)5-s + (−0.919 − 0.391i)7-s + (−0.589 + 0.807i)9-s + (0.354 + 0.935i)11-s + (0.909 + 0.416i)13-s + (0.830 + 0.556i)15-s + (0.711 + 0.702i)17-s + (0.0670 + 0.997i)21-s + (−0.173 − 0.984i)23-s + (0.523 − 0.852i)25-s + (0.987 + 0.160i)27-s + (0.897 − 0.440i)29-s + (0.0402 − 0.999i)31-s + (0.673 − 0.739i)33-s + ⋯
L(s)  = 1  + (−0.452 − 0.891i)3-s + (−0.872 + 0.488i)5-s + (−0.919 − 0.391i)7-s + (−0.589 + 0.807i)9-s + (0.354 + 0.935i)11-s + (0.909 + 0.416i)13-s + (0.830 + 0.556i)15-s + (0.711 + 0.702i)17-s + (0.0670 + 0.997i)21-s + (−0.173 − 0.984i)23-s + (0.523 − 0.852i)25-s + (0.987 + 0.160i)27-s + (0.897 − 0.440i)29-s + (0.0402 − 0.999i)31-s + (0.673 − 0.739i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $0.999 + 0.0426i$
Analytic conductor: \(27.8824\)
Root analytic conductor: \(27.8824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6004} (1507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6004,\ (0:\ ),\ 0.999 + 0.0426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056088628 + 0.02250588989i\)
\(L(\frac12)\) \(\approx\) \(1.056088628 + 0.02250588989i\)
\(L(1)\) \(\approx\) \(0.7654592943 - 0.08541653812i\)
\(L(1)\) \(\approx\) \(0.7654592943 - 0.08541653812i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.452 - 0.891i)T \)
5 \( 1 + (-0.872 + 0.488i)T \)
7 \( 1 + (-0.919 - 0.391i)T \)
11 \( 1 + (0.354 + 0.935i)T \)
13 \( 1 + (0.909 + 0.416i)T \)
17 \( 1 + (0.711 + 0.702i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.897 - 0.440i)T \)
31 \( 1 + (0.0402 - 0.999i)T \)
37 \( 1 + (0.200 + 0.979i)T \)
41 \( 1 + (0.872 - 0.488i)T \)
43 \( 1 + (0.859 + 0.511i)T \)
47 \( 1 + (-0.999 - 0.0268i)T \)
53 \( 1 + (-0.998 + 0.0536i)T \)
59 \( 1 + (-0.0938 + 0.995i)T \)
61 \( 1 + (-0.930 + 0.367i)T \)
67 \( 1 + (0.303 - 0.952i)T \)
71 \( 1 + (0.226 + 0.974i)T \)
73 \( 1 + (0.909 - 0.416i)T \)
83 \( 1 + (-0.428 - 0.903i)T \)
89 \( 1 + (0.730 + 0.682i)T \)
97 \( 1 + (0.523 + 0.852i)T \)
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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

−17.63624598599410428093835187481, −16.77798800939100912000852813328, −16.15986580093184057931720558421, −15.89850882366267515020333888188, −15.511989809878639232396116170255, −14.45806850991330674836901362595, −13.915454846863170005139273276560, −12.903479974550087193460422914031, −12.366024192280350034661123235237, −11.70264328968637382403043075663, −11.12179132137497903526246887713, −10.55078541227007256555605803266, −9.57069344334412432273721416317, −9.16453069591172469461844115539, −8.49172650825870144271120319855, −7.79952104862609255760926184536, −6.764499804631779638769017759795, −6.06307440327826284531665673956, −5.466903975096705898102431440398, −4.80977639785280779545728224465, −3.79903275068537589102136829134, −3.420613639166697123302045348098, −2.88692062324947430520318947368, −1.20895511749789404595382522852, −0.52197595292910134030989163829, 0.64006285964963921676430552753, 1.4470748707326029106829999228, 2.470395353184992659672260797919, 3.1752615509739035289192933981, 4.099140468235978143199289798770, 4.53565902039410062602392508181, 5.84042773261426164425471645784, 6.49772744841022312244487145140, 6.708311347043784669316833580423, 7.728456856460577899120851945968, 7.99691071657099799585657790828, 8.98003953455716463165151772258, 9.89367227841444711995607043700, 10.56814169280051792383783785630, 11.152241144282592177557561583301, 11.95156473438363901939652560504, 12.3984474749345193315880429414, 12.960621330897306068247325666923, 13.74842863578069924977732111264, 14.38140846366915240091763004478, 15.09573061549856598115022136525, 15.89029364275890176851762719430, 16.46318212121036527778941537709, 17.02961768144599823073760057604, 17.79161182486224312244622885066

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