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Properties

Label 1-6008-6008.93-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.356 - 0.934i$
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  + (0.285 + 0.958i)3-s + (−0.617 − 0.786i)5-s + (−0.332 + 0.942i)7-s + (−0.837 + 0.546i)9-s + (0.637 − 0.770i)11-s + (0.675 + 0.737i)13-s + (0.577 − 0.816i)15-s + (−0.745 − 0.666i)17-s + (0.470 + 0.882i)19-s + (−0.998 − 0.0502i)21-s + (−0.984 − 0.175i)23-s + (−0.236 + 0.971i)25-s + (−0.762 − 0.647i)27-s + (−0.260 − 0.965i)29-s + (0.998 + 0.0502i)31-s + ⋯
L(s)  = 1  + (0.285 + 0.958i)3-s + (−0.617 − 0.786i)5-s + (−0.332 + 0.942i)7-s + (−0.837 + 0.546i)9-s + (0.637 − 0.770i)11-s + (0.675 + 0.737i)13-s + (0.577 − 0.816i)15-s + (−0.745 − 0.666i)17-s + (0.470 + 0.882i)19-s + (−0.998 − 0.0502i)21-s + (−0.984 − 0.175i)23-s + (−0.236 + 0.971i)25-s + (−0.762 − 0.647i)27-s + (−0.260 − 0.965i)29-s + (0.998 + 0.0502i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.356 - 0.934i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.356 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5712766426 - 0.3934493542i\)
\(L(\frac12)\) \(\approx\) \(0.5712766426 - 0.3934493542i\)
\(L(1)\) \(\approx\) \(0.8647659468 + 0.1934646947i\)
\(L(1)\) \(\approx\) \(0.8647659468 + 0.1934646947i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.285 + 0.958i)T \)
5 \( 1 + (-0.617 - 0.786i)T \)
7 \( 1 + (-0.332 + 0.942i)T \)
11 \( 1 + (0.637 - 0.770i)T \)
13 \( 1 + (0.675 + 0.737i)T \)
17 \( 1 + (-0.745 - 0.666i)T \)
19 \( 1 + (0.470 + 0.882i)T \)
23 \( 1 + (-0.984 - 0.175i)T \)
29 \( 1 + (-0.260 - 0.965i)T \)
31 \( 1 + (0.998 + 0.0502i)T \)
37 \( 1 + (0.997 - 0.0753i)T \)
41 \( 1 + (-0.637 + 0.770i)T \)
43 \( 1 + (0.0376 - 0.999i)T \)
47 \( 1 + (-0.962 - 0.272i)T \)
53 \( 1 + (-0.0627 - 0.998i)T \)
59 \( 1 + (-0.448 + 0.893i)T \)
61 \( 1 + (-0.0627 - 0.998i)T \)
67 \( 1 + (-0.356 + 0.934i)T \)
71 \( 1 + (-0.597 + 0.801i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.470 + 0.882i)T \)
83 \( 1 + (0.425 - 0.904i)T \)
89 \( 1 + (-0.0376 - 0.999i)T \)
97 \( 1 + (-0.332 - 0.942i)T \)
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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

−17.87423168887329124716550995548, −17.48113115340746150902473070110, −16.591150716163458712697791578440, −15.74550650931429323062591695699, −15.12283249273950926536693803603, −14.56384826060035715518647292287, −13.74754613309629656188019097237, −13.3985041853766028361102192651, −12.57785754062596219793533094024, −12.00880395119382482144209651101, −11.16763377625203310071529065048, −10.765604178752264218375341433353, −9.88216961577811406251213263663, −9.14340753180153515124700704348, −8.1863107207832643656105064354, −7.74962584898884113682155051451, −7.057308040584277596794337519945, −6.511932578315080562455317199771, −6.06480933692714601128070507284, −4.728970530034756610440688710972, −3.96178966865413960531270158322, −3.33077212726685293703257414046, −2.67244755269165766082800770856, −1.68104061858300823541634118485, −0.91996896184644339531210794581, 0.19127575400699645524891168627, 1.46250593316034952433354233969, 2.39991116237449080738456284043, 3.27591078675888140482702870021, 3.93225141768241238500965024336, 4.44362144611517445084202511531, 5.30095625588269803978673358896, 5.96908940622934794646271635507, 6.59011209023392843034211528341, 7.90659262683288905649858379601, 8.39862921832654249759910795749, 8.897850507127989748426584799927, 9.52113255288653762290423662513, 10.0477629102901796592739956842, 11.23996573600246328852156382184, 11.65951340908206418326263697298, 11.98319407438643267017670378523, 13.12440586815050913754051251695, 13.69722838534255674734533816986, 14.37609401961786238898673858949, 15.14537593347183589280461438166, 15.79969801273612085327827038539, 16.187933308187233003625679825756, 16.59422856786695746979061220236, 17.393801391838921464873880845090

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