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Properties

Label 1-5265-5265.3604-r0-0-0
Degree $1$
Conductor $5265$
Sign $0.992 - 0.119i$
Analytic cond. $24.4505$
Root an. cond. $24.4505$
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  + (−0.396 + 0.918i)2-s + (−0.686 − 0.727i)4-s + (0.686 − 0.727i)7-s + (0.939 − 0.342i)8-s + (−0.835 − 0.549i)11-s + (0.396 + 0.918i)14-s + (−0.0581 + 0.998i)16-s + (0.939 + 0.342i)17-s + (0.173 − 0.984i)19-s + (0.835 − 0.549i)22-s + (0.686 + 0.727i)23-s − 28-s + (0.597 + 0.802i)29-s + (−0.686 − 0.727i)31-s + (−0.893 − 0.448i)32-s + ⋯
L(s)  = 1  + (−0.396 + 0.918i)2-s + (−0.686 − 0.727i)4-s + (0.686 − 0.727i)7-s + (0.939 − 0.342i)8-s + (−0.835 − 0.549i)11-s + (0.396 + 0.918i)14-s + (−0.0581 + 0.998i)16-s + (0.939 + 0.342i)17-s + (0.173 − 0.984i)19-s + (0.835 − 0.549i)22-s + (0.686 + 0.727i)23-s − 28-s + (0.597 + 0.802i)29-s + (−0.686 − 0.727i)31-s + (−0.893 − 0.448i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5265\)    =    \(3^{4} \cdot 5 \cdot 13\)
Sign: $0.992 - 0.119i$
Analytic conductor: \(24.4505\)
Root analytic conductor: \(24.4505\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5265} (3604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5265,\ (0:\ ),\ 0.992 - 0.119i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.486924552 - 0.08905386722i\)
\(L(\frac12)\) \(\approx\) \(1.486924552 - 0.08905386722i\)
\(L(1)\) \(\approx\) \(0.9217611443 + 0.1694987177i\)
\(L(1)\) \(\approx\) \(0.9217611443 + 0.1694987177i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.396 + 0.918i)T \)
7 \( 1 + (0.686 - 0.727i)T \)
11 \( 1 + (-0.835 - 0.549i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.686 + 0.727i)T \)
29 \( 1 + (0.597 + 0.802i)T \)
31 \( 1 + (-0.686 - 0.727i)T \)
37 \( 1 + (0.939 + 0.342i)T \)
41 \( 1 + (0.597 - 0.802i)T \)
43 \( 1 + (0.0581 - 0.998i)T \)
47 \( 1 + (0.686 - 0.727i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.835 + 0.549i)T \)
61 \( 1 + (-0.286 - 0.957i)T \)
67 \( 1 + (0.993 + 0.116i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (-0.993 + 0.116i)T \)
83 \( 1 + (0.993 - 0.116i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.835 + 0.549i)T \)
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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

−18.14773598100517573934809236628, −17.615216067788257941675303881671, −16.67994320560996339511291214600, −16.208240056750742441120713867730, −15.256931465420011891614575534558, −14.504066431509253220711836063003, −14.04128980778995459305475287210, −12.98134616264123376825877398188, −12.52395572743222753353313199714, −11.989725452635098551943640074075, −11.22321241063822051397917514399, −10.652616095509874599279506607792, −9.8835077244007590125495806842, −9.377387715351643107524623397, −8.51268793561900895702468726155, −7.87439853326340441811038121492, −7.50458862850344889548539660410, −6.24390581377350797015823505193, −5.324858371814395625586043904693, −4.811542327845974142505507963586, −4.025668878628078156268048710338, −2.97474449012165448472358309029, −2.52265990139214129951311048938, −1.66756173084723860011167624433, −0.87386754290424671986431781758, 0.61468607734374817119826574868, 1.231067407738495416935235607586, 2.350278060497674192560825386498, 3.45262725130747652894464810079, 4.21398847996071273152239865940, 5.170313248810982335747994327033, 5.42337372222403045863830100717, 6.38538092607268692732362202815, 7.334109513994684493972034223795, 7.564927514918179659605332203194, 8.35699379669013737489043891889, 9.020588419257049354495595046773, 9.757643873698122881431357321977, 10.7469813532427034785066262274, 10.782684430818266171091606953357, 11.86295189903941310901543101770, 12.93414593850177377525324664488, 13.465944526818367979105624109693, 14.06811138651624122836382578858, 14.68008125624732764737685786527, 15.423237547016992332915259043029, 15.91173209094154119423057480071, 16.84688093282718319014968700857, 17.07827677773380602592146895988, 17.84123781987691438184503364064

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