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Properties

Label 1-4011-4011.551-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.0916 + 0.995i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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  + (0.968 + 0.250i)2-s + (0.874 + 0.485i)4-s + (−0.754 − 0.656i)5-s + (0.724 + 0.689i)8-s + (−0.565 − 0.824i)10-s + (0.962 + 0.272i)11-s + (−0.965 − 0.261i)13-s + (0.528 + 0.849i)16-s + (0.685 − 0.728i)17-s + (−0.565 + 0.824i)19-s + (−0.340 − 0.940i)20-s + (0.863 + 0.504i)22-s + (0.609 + 0.792i)23-s + (0.137 + 0.990i)25-s + (−0.868 − 0.495i)26-s + ⋯
L(s)  = 1  + (0.968 + 0.250i)2-s + (0.874 + 0.485i)4-s + (−0.754 − 0.656i)5-s + (0.724 + 0.689i)8-s + (−0.565 − 0.824i)10-s + (0.962 + 0.272i)11-s + (−0.965 − 0.261i)13-s + (0.528 + 0.849i)16-s + (0.685 − 0.728i)17-s + (−0.565 + 0.824i)19-s + (−0.340 − 0.940i)20-s + (0.863 + 0.504i)22-s + (0.609 + 0.792i)23-s + (0.137 + 0.990i)25-s + (−0.868 − 0.495i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0916 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0916 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.0916 + 0.995i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.0916 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.639381587 + 1.797264054i\)
\(L(\frac12)\) \(\approx\) \(1.639381587 + 1.797264054i\)
\(L(1)\) \(\approx\) \(1.587340144 + 0.4072183838i\)
\(L(1)\) \(\approx\) \(1.587340144 + 0.4072183838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.968 + 0.250i)T \)
5 \( 1 + (-0.754 - 0.656i)T \)
11 \( 1 + (0.962 + 0.272i)T \)
13 \( 1 + (-0.965 - 0.261i)T \)
17 \( 1 + (0.685 - 0.728i)T \)
19 \( 1 + (-0.565 + 0.824i)T \)
23 \( 1 + (0.609 + 0.792i)T \)
29 \( 1 + (-0.828 + 0.560i)T \)
31 \( 1 + (0.592 + 0.805i)T \)
37 \( 1 + (-0.592 + 0.805i)T \)
41 \( 1 + (-0.986 - 0.164i)T \)
43 \( 1 + (-0.909 + 0.416i)T \)
47 \( 1 + (-0.441 - 0.897i)T \)
53 \( 1 + (-0.938 + 0.345i)T \)
59 \( 1 + (0.952 + 0.303i)T \)
61 \( 1 + (0.982 + 0.186i)T \)
67 \( 1 + (-0.795 + 0.605i)T \)
71 \( 1 + (0.461 - 0.887i)T \)
73 \( 1 + (-0.884 + 0.466i)T \)
79 \( 1 + (0.0715 - 0.997i)T \)
83 \( 1 + (0.991 + 0.131i)T \)
89 \( 1 + (0.509 - 0.860i)T \)
97 \( 1 + (-0.746 + 0.665i)T \)
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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

−18.7125626983396108723049657654, −17.398417754482076567758508705162, −16.86552175437474876600532060011, −16.14057203293392551487239191844, −15.23553063179860947687548296914, −14.792346013794831061785098670439, −14.42483103822253408361634334022, −13.544369464926728876509429872, −12.73429680985355162515610857591, −12.11747489309074229279418157129, −11.50176702679235409845049490412, −10.98668445721171175844827221003, −10.201687107691056367868912305568, −9.48308512753552317462340997187, −8.406235904993498859344574772668, −7.6239776355386844542847236283, −6.7535019792132663120508795431, −6.50995850335192685753154166751, −5.46094717905955180777113629178, −4.590074265760939269249628311331, −3.987098498407697295183751408433, −3.317534268798511089503099338689, −2.52708837980553384073540244239, −1.73004651174010232649904170762, −0.46787886919685498854012044090, 1.20127199694442971457114903754, 1.94291754884074286028281679024, 3.28710827786191313237894537968, 3.514514716600786313030005598842, 4.63623496735410375576259550574, 4.98712213743842579546356472702, 5.77317024949868096068278811745, 6.86772995578777731202045890283, 7.24201146241330099008557024343, 8.0728877751592578481920360769, 8.74013783333890656515167150088, 9.70099474912940235183185818725, 10.463428719158923019088429360002, 11.66795590867278660029468934136, 11.77902968445100340534073581129, 12.48287889689282383130260553830, 13.12424702895150161768841772204, 13.918753513425895094939031323957, 14.84269090548672672800016184025, 14.94039518161083203793295396012, 15.91164563653870988736786284252, 16.60524987182287385993655692738, 16.96749904679131746638962104311, 17.69792599090610636988461533709, 18.99646829956292400713108037282

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