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Properties

Label 1-6008-6008.1029-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.0979 + 0.995i$
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  + (0.788 + 0.614i)3-s + (0.987 + 0.158i)5-s + (0.899 + 0.437i)7-s + (0.244 + 0.969i)9-s + (−0.387 + 0.921i)11-s + (0.998 + 0.0586i)13-s + (0.681 + 0.731i)15-s + (0.563 + 0.825i)17-s + (−0.129 − 0.991i)19-s + (0.440 + 0.897i)21-s + (0.972 + 0.232i)23-s + (0.949 + 0.312i)25-s + (−0.402 + 0.915i)27-s + (0.170 − 0.985i)29-s + (0.997 + 0.0669i)31-s + ⋯
L(s)  = 1  + (0.788 + 0.614i)3-s + (0.987 + 0.158i)5-s + (0.899 + 0.437i)7-s + (0.244 + 0.969i)9-s + (−0.387 + 0.921i)11-s + (0.998 + 0.0586i)13-s + (0.681 + 0.731i)15-s + (0.563 + 0.825i)17-s + (−0.129 − 0.991i)19-s + (0.440 + 0.897i)21-s + (0.972 + 0.232i)23-s + (0.949 + 0.312i)25-s + (−0.402 + 0.915i)27-s + (0.170 − 0.985i)29-s + (0.997 + 0.0669i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0979 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0979 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.0979 + 0.995i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.0979 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.045747949 + 2.760593178i\)
\(L(\frac12)\) \(\approx\) \(3.045747949 + 2.760593178i\)
\(L(1)\) \(\approx\) \(1.842651797 + 0.7832837785i\)
\(L(1)\) \(\approx\) \(1.842651797 + 0.7832837785i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.788 + 0.614i)T \)
5 \( 1 + (0.987 + 0.158i)T \)
7 \( 1 + (0.899 + 0.437i)T \)
11 \( 1 + (-0.387 + 0.921i)T \)
13 \( 1 + (0.998 + 0.0586i)T \)
17 \( 1 + (0.563 + 0.825i)T \)
19 \( 1 + (-0.129 - 0.991i)T \)
23 \( 1 + (0.972 + 0.232i)T \)
29 \( 1 + (0.170 - 0.985i)T \)
31 \( 1 + (0.997 + 0.0669i)T \)
37 \( 1 + (0.584 + 0.811i)T \)
41 \( 1 + (-0.992 + 0.125i)T \)
43 \( 1 + (-0.998 - 0.0502i)T \)
47 \( 1 + (-0.154 - 0.988i)T \)
53 \( 1 + (0.425 - 0.904i)T \)
59 \( 1 + (-0.0961 + 0.995i)T \)
61 \( 1 + (-0.996 + 0.0836i)T \)
67 \( 1 + (-0.884 - 0.467i)T \)
71 \( 1 + (0.656 + 0.754i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.129 - 0.991i)T \)
83 \( 1 + (0.895 - 0.444i)T \)
89 \( 1 + (-0.542 - 0.839i)T \)
97 \( 1 + (-0.828 - 0.560i)T \)
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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

−17.80398798550830568556916688725, −16.83168556654887329911112320615, −16.47315907856781038295558548707, −15.48284796202185752181474305209, −14.6888553037691872181491076803, −14.06820315668410417117740015745, −13.75825275773624884235233224806, −13.18810806603623971831599551020, −12.45897805009925637551968476656, −11.67094122173392953355818605736, −10.81292548826028497037912311898, −10.341413556709727972463102527034, −9.38751413306601854686924995152, −8.80755781825464213208380899359, −8.18662176621495794131897872284, −7.678667407866141193018412747108, −6.737846001395708383513095254249, −6.14111590711775195147424731280, −5.375900236445925009156328068467, −4.66267263770276825653401888901, −3.5642248308276386697560367587, −3.020322674236870256720686297595, −2.18369818602068622235872359051, −1.23150103698647450188713000869, −1.024993927643477452789321861191, 1.326971956795303973916958810145, 1.85010964861028155011615900070, 2.63146855484908609908924280224, 3.23803305861727316972224784542, 4.33496901839686232356951202427, 4.86593162140534103788067679250, 5.49415547562702213030370884620, 6.35476886807159823075252918770, 7.16490743678962652507839657921, 8.061362785957016370440299408495, 8.586770570605791273643217647, 9.13319592892665056272845962853, 10.08145373482800173340730048504, 10.26076848770850614351554177389, 11.170003607162896211178979258447, 11.80053816466007295829204435910, 12.92635951054306212470721926293, 13.412808353631790587492743015613, 13.86853849269043699472123121449, 14.84452118257115585823944062939, 15.115754942329699066037380093060, 15.56779816813407509915246019524, 16.71827928780103409717557824624, 17.153882026921847812914148781303, 17.92702529626993712133298745691

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