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Properties

Label 1-6008-6008.965-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.971 - 0.235i$
Analytic cond. $27.9010$
Root an. cond. $27.9010$
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  + (0.0544 + 0.998i)3-s + (0.723 + 0.690i)5-s + (−0.778 − 0.627i)7-s + (−0.994 + 0.108i)9-s + (−0.944 − 0.328i)11-s + (0.818 + 0.574i)13-s + (−0.650 + 0.759i)15-s + (0.804 − 0.594i)17-s + (0.440 + 0.897i)19-s + (0.584 − 0.811i)21-s + (−0.767 + 0.640i)23-s + (0.0460 + 0.998i)25-s + (−0.162 − 0.986i)27-s + (0.00418 + 0.999i)29-s + (−0.410 − 0.911i)31-s + ⋯
L(s)  = 1  + (0.0544 + 0.998i)3-s + (0.723 + 0.690i)5-s + (−0.778 − 0.627i)7-s + (−0.994 + 0.108i)9-s + (−0.944 − 0.328i)11-s + (0.818 + 0.574i)13-s + (−0.650 + 0.759i)15-s + (0.804 − 0.594i)17-s + (0.440 + 0.897i)19-s + (0.584 − 0.811i)21-s + (−0.767 + 0.640i)23-s + (0.0460 + 0.998i)25-s + (−0.162 − 0.986i)27-s + (0.00418 + 0.999i)29-s + (−0.410 − 0.911i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.971 - 0.235i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1150580560 + 0.9631481696i\)
\(L(\frac12)\) \(\approx\) \(-0.1150580560 + 0.9631481696i\)
\(L(1)\) \(\approx\) \(0.8472043766 + 0.4755550673i\)
\(L(1)\) \(\approx\) \(0.8472043766 + 0.4755550673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.0544 + 0.998i)T \)
5 \( 1 + (0.723 + 0.690i)T \)
7 \( 1 + (-0.778 - 0.627i)T \)
11 \( 1 + (-0.944 - 0.328i)T \)
13 \( 1 + (0.818 + 0.574i)T \)
17 \( 1 + (0.804 - 0.594i)T \)
19 \( 1 + (0.440 + 0.897i)T \)
23 \( 1 + (-0.767 + 0.640i)T \)
29 \( 1 + (0.00418 + 0.999i)T \)
31 \( 1 + (-0.410 - 0.911i)T \)
37 \( 1 + (0.624 + 0.781i)T \)
41 \( 1 + (-0.187 - 0.982i)T \)
43 \( 1 + (0.997 + 0.0753i)T \)
47 \( 1 + (-0.880 + 0.474i)T \)
53 \( 1 + (0.992 - 0.125i)T \)
59 \( 1 + (-0.993 + 0.117i)T \)
61 \( 1 + (-0.604 - 0.796i)T \)
67 \( 1 + (-0.949 + 0.312i)T \)
71 \( 1 + (-0.285 - 0.958i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.440 + 0.897i)T \)
83 \( 1 + (0.348 - 0.937i)T \)
89 \( 1 + (0.563 + 0.825i)T \)
97 \( 1 + (0.932 + 0.360i)T \)
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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

−17.733213744393257843731342532827, −16.60148533049890200963870110789, −16.23741892142528952695898733995, −15.42704841049539405668332992721, −14.70359513108508536875910268795, −13.78569186015656792402095586409, −13.29315414533496292436230677633, −12.81231995199284021346487748311, −12.35623678538809616995323626016, −11.664246854586775260341950801535, −10.631433982766199556937963764724, −10.050934069837236113422305150892, −9.22752576098355232793279240228, −8.63683491260938391272194451956, −8.01281896417205255368745577886, −7.3403741943595428505154717146, −6.32620400853470889930099516382, −5.89066474145314398146666307114, −5.4378978633535638224887704908, −4.48473877884257229555885860988, −3.27726588691340019744932138219, −2.68267101618904078851324413967, −1.99781980566779149173496639512, −1.14764110144574539242953785927, −0.24896197154861519671009688683, 1.174075561495659531337055534437, 2.24724896264523588054601968364, 3.18600313369310606560744147748, 3.43740010705198150518724537042, 4.277897355074985605976645509431, 5.31227424570960930512736799421, 5.84382996114909792779779230362, 6.37146913740084246879902115191, 7.42777881525428234730897960979, 7.92820468126565718164708090596, 9.064713861860964833151693682121, 9.51530183385781331201841362794, 10.21420425021911418584222266936, 10.538409673682135825665617508011, 11.2552888210568728281112426747, 11.98809364597103480197842879386, 13.05949649516894385916052228522, 13.659682630046066542067381495175, 14.11686633089481017211080547225, 14.690483765291310173811652563106, 15.66272257640804380752105190814, 16.10004483521592080983632263833, 16.61579887849918312418042619681, 17.223613921826082473871101757758, 18.25232113831694597983118104234

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