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Properties

Label 1-6004-6004.359-r0-0-0
Degree $1$
Conductor $6004$
Sign $0.866 + 0.498i$
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  + (−0.589 + 0.807i)3-s + (0.523 − 0.852i)5-s + (0.692 + 0.721i)7-s + (−0.303 − 0.952i)9-s + (0.748 − 0.663i)11-s + (0.653 + 0.757i)13-s + (0.379 + 0.925i)15-s + (−0.0134 − 0.999i)17-s + (−0.991 + 0.133i)21-s + (0.939 − 0.342i)23-s + (−0.452 − 0.891i)25-s + (0.948 + 0.316i)27-s + (−0.611 + 0.791i)29-s + (0.996 + 0.0804i)31-s + (0.0938 + 0.995i)33-s + ⋯
L(s)  = 1  + (−0.589 + 0.807i)3-s + (0.523 − 0.852i)5-s + (0.692 + 0.721i)7-s + (−0.303 − 0.952i)9-s + (0.748 − 0.663i)11-s + (0.653 + 0.757i)13-s + (0.379 + 0.925i)15-s + (−0.0134 − 0.999i)17-s + (−0.991 + 0.133i)21-s + (0.939 − 0.342i)23-s + (−0.452 − 0.891i)25-s + (0.948 + 0.316i)27-s + (−0.611 + 0.791i)29-s + (0.996 + 0.0804i)31-s + (0.0938 + 0.995i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $0.866 + 0.498i$
Analytic conductor: \(27.8824\)
Root analytic conductor: \(27.8824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6004} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6004,\ (0:\ ),\ 0.866 + 0.498i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.068308799 + 0.5527697323i\)
\(L(\frac12)\) \(\approx\) \(2.068308799 + 0.5527697323i\)
\(L(1)\) \(\approx\) \(1.186692428 + 0.1760018793i\)
\(L(1)\) \(\approx\) \(1.186692428 + 0.1760018793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.589 + 0.807i)T \)
5 \( 1 + (0.523 - 0.852i)T \)
7 \( 1 + (0.692 + 0.721i)T \)
11 \( 1 + (0.748 - 0.663i)T \)
13 \( 1 + (0.653 + 0.757i)T \)
17 \( 1 + (-0.0134 - 0.999i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.611 + 0.791i)T \)
31 \( 1 + (0.996 + 0.0804i)T \)
37 \( 1 + (0.919 - 0.391i)T \)
41 \( 1 + (-0.523 + 0.852i)T \)
43 \( 1 + (0.476 + 0.879i)T \)
47 \( 1 + (0.998 + 0.0536i)T \)
53 \( 1 + (-0.994 + 0.107i)T \)
59 \( 1 + (-0.982 - 0.186i)T \)
61 \( 1 + (-0.730 + 0.682i)T \)
67 \( 1 + (0.815 + 0.579i)T \)
71 \( 1 + (-0.897 + 0.440i)T \)
73 \( 1 + (0.653 - 0.757i)T \)
83 \( 1 + (0.632 - 0.774i)T \)
89 \( 1 + (0.0670 + 0.997i)T \)
97 \( 1 + (-0.452 + 0.891i)T \)
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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

−17.54397603254729208607401437155, −17.14204149960700903065989539437, −16.87547980062297089387674234703, −15.36259140051807602886400567412, −15.18986981860449402002806605343, −14.10526129609086243873892234451, −13.83575992150628896242288803001, −13.0694006916792824330061642739, −12.47498581919665410572581643054, −11.57011617509826666539166970541, −11.0639662270285807464472114952, −10.53697695007919079267318792549, −9.93197896458800908059526430307, −8.933383315821851071527237715611, −8.01083642993977955270011436614, −7.51864078613651122897868962763, −6.8459186574264020678237526129, −6.21709378777667720157117224301, −5.672247975230123894159967642046, −4.77035014032187886567385556880, −3.95945244537993106982278503838, −3.10600888306333190437952522622, −2.080418979316346363044950752098, −1.55786651867395249578638614627, −0.76222875824895356751178563452, 0.89347530731850503251441222776, 1.395098139500422542563385814018, 2.54589745672465675704111686735, 3.37191484052695996122645499155, 4.457537403848482713832035793515, 4.69913207479641792457561545817, 5.524935153210994767331276989679, 6.10144730067192522924887928153, 6.68974687880768039958042814898, 7.90556704060832376195836892679, 8.75614045933378022446893090089, 9.180238639802470368033352117288, 9.49219021827870203674933791571, 10.63274679939020130811213919417, 11.244422165565830876730788734811, 11.72412477915954845992274444358, 12.30914511324509359610882755961, 13.16013454697666706231211348165, 13.98211439730898859653552988963, 14.47929832028159647746134559719, 15.23353318942093664668865906492, 16.058740268452728712542612634103, 16.410209886854458999508745248144, 17.02157131772029590883468453159, 17.62345003483675810668587804951

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