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Properties

Label 1-6008-6008.3875-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.885 - 0.464i$
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  + (−0.604 − 0.796i)3-s + (−0.944 + 0.328i)5-s + (−0.992 + 0.125i)7-s + (−0.268 + 0.963i)9-s + (−0.669 + 0.743i)11-s + (0.999 + 0.0418i)13-s + (0.832 + 0.553i)15-s + (0.0209 − 0.999i)17-s + (−0.348 + 0.937i)19-s + (0.699 + 0.714i)21-s + (−0.985 − 0.166i)23-s + (0.783 − 0.621i)25-s + (0.929 − 0.368i)27-s + (−0.699 − 0.714i)29-s + (−0.268 + 0.963i)31-s + ⋯
L(s)  = 1  + (−0.604 − 0.796i)3-s + (−0.944 + 0.328i)5-s + (−0.992 + 0.125i)7-s + (−0.268 + 0.963i)9-s + (−0.669 + 0.743i)11-s + (0.999 + 0.0418i)13-s + (0.832 + 0.553i)15-s + (0.0209 − 0.999i)17-s + (−0.348 + 0.937i)19-s + (0.699 + 0.714i)21-s + (−0.985 − 0.166i)23-s + (0.783 − 0.621i)25-s + (0.929 − 0.368i)27-s + (−0.699 − 0.714i)29-s + (−0.268 + 0.963i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.885 - 0.464i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (3875, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.885 - 0.464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3131533357 - 0.07715881927i\)
\(L(\frac12)\) \(\approx\) \(0.3131533357 - 0.07715881927i\)
\(L(1)\) \(\approx\) \(0.5080793557 - 0.05256677068i\)
\(L(1)\) \(\approx\) \(0.5080793557 - 0.05256677068i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.604 - 0.796i)T \)
5 \( 1 + (-0.944 + 0.328i)T \)
7 \( 1 + (-0.992 + 0.125i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.999 + 0.0418i)T \)
17 \( 1 + (0.0209 - 0.999i)T \)
19 \( 1 + (-0.348 + 0.937i)T \)
23 \( 1 + (-0.985 - 0.166i)T \)
29 \( 1 + (-0.699 - 0.714i)T \)
31 \( 1 + (-0.268 + 0.963i)T \)
37 \( 1 + (-0.783 + 0.621i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (-0.187 - 0.982i)T \)
47 \( 1 + (-0.944 + 0.328i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.957 - 0.289i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (0.699 + 0.714i)T \)
71 \( 1 + (-0.0627 - 0.998i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.348 - 0.937i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (-0.756 - 0.653i)T \)
97 \( 1 + (0.604 - 0.796i)T \)
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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

−17.71705542056553925471627695256, −16.77426876205381985440169578, −16.35730829287478250570497043580, −15.95302960824406034569151987105, −15.32020579960473623576840446451, −14.84038002486876573792366218027, −13.69312582744074598027613640052, −12.99294881476217799788054550125, −12.5861425568155043395611487689, −11.65254522164764969913604608891, −11.05349014573647174547256183700, −10.68484153572855108388143093936, −9.83685079516144478416388674854, −9.11819091128387974381253782534, −8.470480102131577338641675767273, −7.858147320377079131467051463759, −6.79667577878223300532293171708, −6.186586348869041110381015128520, −5.58437453054892359717281317747, −4.76241866148524259535589189555, −3.87408728731910658544087583797, −3.59702664042590073580966843028, −2.84506606339734229561997850490, −1.42319488130776680220625759897, −0.32527750603185793595547473413, 0.26597443582380694142560021979, 1.50447559214887869119502553176, 2.3240136269381796048514339034, 3.19487491031262046317194967190, 3.84999962975856065563997575693, 4.76410897305253018532314827967, 5.58403409256570754081306981253, 6.30186151333859717494672131926, 6.88182077327448486445736179357, 7.505415649599025137857768544719, 8.10725562387820614705898817064, 8.83096759846377768122729283280, 9.89142817664731823908199227506, 10.47715204219473992929527482584, 11.10459348644563296590779471343, 11.93266326372535883025403550282, 12.3378266072156989229003918207, 12.85393760382860133423615507116, 13.72942547632209770217266796863, 14.15610191998313494409791597039, 15.39080633293234592285902661019, 15.69664830778613720247952401122, 16.32697127332124032210914586223, 16.91546651707407133690264161710, 17.86882145358529606953673627533

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