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Properties

Label 1-6027-6027.488-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.823 + 0.566i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.791 + 0.611i)2-s + (0.251 − 0.967i)4-s + (−0.946 − 0.323i)5-s + (0.393 + 0.919i)8-s + (0.946 − 0.323i)10-s + (−0.525 − 0.850i)11-s + (−0.134 − 0.990i)13-s + (−0.873 − 0.486i)16-s + (0.712 − 0.701i)17-s + (−0.669 + 0.743i)19-s + (−0.550 + 0.834i)20-s + (0.936 + 0.351i)22-s + (0.447 − 0.894i)23-s + (0.791 + 0.611i)25-s + (0.712 + 0.701i)26-s + ⋯
L(s)  = 1  + (−0.791 + 0.611i)2-s + (0.251 − 0.967i)4-s + (−0.946 − 0.323i)5-s + (0.393 + 0.919i)8-s + (0.946 − 0.323i)10-s + (−0.525 − 0.850i)11-s + (−0.134 − 0.990i)13-s + (−0.873 − 0.486i)16-s + (0.712 − 0.701i)17-s + (−0.669 + 0.743i)19-s + (−0.550 + 0.834i)20-s + (0.936 + 0.351i)22-s + (0.447 − 0.894i)23-s + (0.791 + 0.611i)25-s + (0.712 + 0.701i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.823 + 0.566i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.823 + 0.566i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01422709712 + 0.04579976268i\)
\(L(\frac12)\) \(\approx\) \(0.01422709712 + 0.04579976268i\)
\(L(1)\) \(\approx\) \(0.5255171173 + 0.009809772020i\)
\(L(1)\) \(\approx\) \(0.5255171173 + 0.009809772020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.791 + 0.611i)T \)
5 \( 1 + (-0.946 - 0.323i)T \)
11 \( 1 + (-0.525 - 0.850i)T \)
13 \( 1 + (-0.134 - 0.990i)T \)
17 \( 1 + (0.712 - 0.701i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (0.447 - 0.894i)T \)
29 \( 1 + (0.0448 + 0.998i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.842 + 0.538i)T \)
43 \( 1 + (-0.995 - 0.0896i)T \)
47 \( 1 + (0.791 - 0.611i)T \)
53 \( 1 + (-0.251 + 0.967i)T \)
59 \( 1 + (0.575 - 0.817i)T \)
61 \( 1 + (-0.998 - 0.0598i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (0.0448 - 0.998i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.925 + 0.379i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−17.264037793386491501699334324957, −17.14236778457777282457372164806, −16.173024661016158042328558890350, −15.51306673001305716849195368945, −15.08951678042349698039617991893, −14.21248157702887373323152914377, −13.20039466168295479888238106748, −12.73074211511300896551734473737, −11.89545123147589281075890945382, −11.53775743300711026093571270972, −10.84432606981332662902850231940, −10.13791100566981424412634014222, −9.56958976882403697775986150503, −8.74737192594660712921892370732, −8.13525697944576332612804204625, −7.38788886917332714736262310212, −7.05535483660905733652524631979, −6.15364427766749987771362184444, −4.938493004742555467633302646, −4.15886536199846987177464415789, −3.713419516425561059207396869871, −2.714114820409249187847320175951, −2.130091260556318745044823281973, −1.218967392780335984720472172415, −0.02264637541113417571867858222, 0.762285057444755670359155967675, 1.59854332258847875973365688108, 2.900381469188768036263015287408, 3.34275252364323507187210075391, 4.57720223362265324641689559161, 5.19915032583731602059279999179, 5.76563071394003831874185497433, 6.75131256735505150099911403780, 7.32038114831380828497766350116, 8.12627775512216159362690243612, 8.42939445668383622473751800748, 9.04265462897092395304339760860, 10.13243702518981827696352679625, 10.56494604031278537140161277340, 11.13810671972261432000848750699, 12.073577720671905191693222189552, 12.56187074787605088048066728378, 13.488662652263517460901968588730, 14.31310408848930707459555816431, 14.83142388016878972933425504574, 15.65158760519395815246151010341, 15.94349703980955758176391564768, 16.78458853376446623324677124804, 16.99677931435754128175775056983, 18.1559493669025289336118340424

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