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Properties

Label 1-6008-6008.853-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.389 + 0.921i$
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  + (−0.999 + 0.0335i)3-s + (−0.891 − 0.452i)5-s + (−0.984 − 0.175i)7-s + (0.997 − 0.0669i)9-s + (−0.387 − 0.921i)11-s + (0.252 − 0.967i)13-s + (0.906 + 0.421i)15-s + (0.0293 + 0.999i)17-s + (0.903 + 0.429i)19-s + (0.989 + 0.141i)21-s + (0.521 + 0.853i)23-s + (0.591 + 0.806i)25-s + (−0.994 + 0.100i)27-s + (−0.884 + 0.467i)29-s + (0.372 + 0.928i)31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0335i)3-s + (−0.891 − 0.452i)5-s + (−0.984 − 0.175i)7-s + (0.997 − 0.0669i)9-s + (−0.387 − 0.921i)11-s + (0.252 − 0.967i)13-s + (0.906 + 0.421i)15-s + (0.0293 + 0.999i)17-s + (0.903 + 0.429i)19-s + (0.989 + 0.141i)21-s + (0.521 + 0.853i)23-s + (0.591 + 0.806i)25-s + (−0.994 + 0.100i)27-s + (−0.884 + 0.467i)29-s + (0.372 + 0.928i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.389 + 0.921i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.389 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1355803200 + 0.2045241769i\)
\(L(\frac12)\) \(\approx\) \(0.1355803200 + 0.2045241769i\)
\(L(1)\) \(\approx\) \(0.5534161927 - 0.05029315920i\)
\(L(1)\) \(\approx\) \(0.5534161927 - 0.05029315920i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.999 + 0.0335i)T \)
5 \( 1 + (-0.891 - 0.452i)T \)
7 \( 1 + (-0.984 - 0.175i)T \)
11 \( 1 + (-0.387 - 0.921i)T \)
13 \( 1 + (0.252 - 0.967i)T \)
17 \( 1 + (0.0293 + 0.999i)T \)
19 \( 1 + (0.903 + 0.429i)T \)
23 \( 1 + (0.521 + 0.853i)T \)
29 \( 1 + (-0.884 + 0.467i)T \)
31 \( 1 + (0.372 + 0.928i)T \)
37 \( 1 + (0.952 + 0.305i)T \)
41 \( 1 + (-0.992 - 0.125i)T \)
43 \( 1 + (0.778 - 0.627i)T \)
47 \( 1 + (0.705 - 0.708i)T \)
53 \( 1 + (0.425 + 0.904i)T \)
59 \( 1 + (-0.507 + 0.861i)T \)
61 \( 1 + (-0.996 - 0.0836i)T \)
67 \( 1 + (-0.717 - 0.696i)T \)
71 \( 1 + (-0.514 - 0.857i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.903 + 0.429i)T \)
83 \( 1 + (0.895 + 0.444i)T \)
89 \( 1 + (0.932 - 0.360i)T \)
97 \( 1 + (0.340 - 0.940i)T \)
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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

−17.527215621665940163168279717616, −16.6780644368382734271071439098, −16.17269440794790796951135466555, −15.7334749857377684663004275409, −15.11140595649773901623879257207, −14.330179378899903072213278330867, −13.26708828440870752852812050535, −12.92666528704039414127177527633, −11.90828455715914967042482302089, −11.77133047167998308542482955478, −11.01582201494176943271119044709, −10.28235465903020021848460498331, −9.55020787763740110382632713321, −9.11558574008208747466285159776, −7.77574060171556173595350675086, −7.30984968277688098851524129565, −6.72721407986660347865434232397, −6.15720723704808709180732485201, −5.228195402830466963697463000326, −4.467701323462864111718903637764, −3.98984776437286473788403266311, −2.937298877452380836158218523611, −2.31008698975854867430625560227, −1.02038077503039860976396619033, −0.11374675245888608480003909460, 0.81060813429223376024462249645, 1.44950041836756713153076152894, 3.10709440263875632954835381843, 3.43830893067226261091719361506, 4.20684838928730563585961356160, 5.172385229938404221379182206947, 5.72692591063992069398081962089, 6.23295560630517285580100245996, 7.34363444820264184215860413030, 7.603772956607225971798262052646, 8.60808163320592573948080265549, 9.24211499215867251891640536505, 10.23888638533920096842810992638, 10.65284828033562260339955099138, 11.26918042654804831300006132983, 12.18563578134486485233955698059, 12.41636408018277903417116509971, 13.33247102095758083637897779045, 13.57163054325859233607330895192, 15.063339375767813004980245238357, 15.440882545733573340221567998184, 16.042032333213107511552463692470, 16.65449493409297223440186388654, 16.97009424676259503096355798618, 17.93455981289593154016477681257

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