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Properties

Label 1-4056-4056.2651-r0-0-0
Degree $1$
Conductor $4056$
Sign $0.389 + 0.921i$
Analytic cond. $18.8359$
Root an. cond. $18.8359$
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  + (−0.885 + 0.464i)5-s + (−0.354 + 0.935i)7-s + (0.120 + 0.992i)11-s + (0.354 − 0.935i)17-s − 19-s + 23-s + (0.568 − 0.822i)25-s + (0.120 − 0.992i)29-s + (0.568 + 0.822i)31-s + (−0.120 − 0.992i)35-s + (0.568 + 0.822i)37-s + (−0.748 + 0.663i)41-s + (0.568 − 0.822i)43-s + (0.970 − 0.239i)47-s + (−0.748 − 0.663i)49-s + ⋯
L(s)  = 1  + (−0.885 + 0.464i)5-s + (−0.354 + 0.935i)7-s + (0.120 + 0.992i)11-s + (0.354 − 0.935i)17-s − 19-s + 23-s + (0.568 − 0.822i)25-s + (0.120 − 0.992i)29-s + (0.568 + 0.822i)31-s + (−0.120 − 0.992i)35-s + (0.568 + 0.822i)37-s + (−0.748 + 0.663i)41-s + (0.568 − 0.822i)43-s + (0.970 − 0.239i)47-s + (−0.748 − 0.663i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4056\)    =    \(2^{3} \cdot 3 \cdot 13^{2}\)
Sign: $0.389 + 0.921i$
Analytic conductor: \(18.8359\)
Root analytic conductor: \(18.8359\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4056} (2651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4056,\ (0:\ ),\ 0.389 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.088617384 + 0.7219151965i\)
\(L(\frac12)\) \(\approx\) \(1.088617384 + 0.7219151965i\)
\(L(1)\) \(\approx\) \(0.8862007425 + 0.2260724081i\)
\(L(1)\) \(\approx\) \(0.8862007425 + 0.2260724081i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.885 + 0.464i)T \)
7 \( 1 + (-0.354 + 0.935i)T \)
11 \( 1 + (0.120 + 0.992i)T \)
17 \( 1 + (0.354 - 0.935i)T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + (0.120 - 0.992i)T \)
31 \( 1 + (0.568 + 0.822i)T \)
37 \( 1 + (0.568 + 0.822i)T \)
41 \( 1 + (-0.748 + 0.663i)T \)
43 \( 1 + (0.568 - 0.822i)T \)
47 \( 1 + (0.970 - 0.239i)T \)
53 \( 1 + (-0.354 + 0.935i)T \)
59 \( 1 + (0.885 - 0.464i)T \)
61 \( 1 + (0.354 + 0.935i)T \)
67 \( 1 + (0.970 - 0.239i)T \)
71 \( 1 + (0.748 - 0.663i)T \)
73 \( 1 + (-0.120 - 0.992i)T \)
79 \( 1 + (0.970 - 0.239i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.885 - 0.464i)T \)
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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

−18.59155041979159321670033736834, −17.18924153662594798154296916392, −17.11924273411814202134143240854, −16.27747397929544940306624427104, −15.799530568886126446860787812612, −14.87388995468517675063121035680, −14.35514185799945065977945504712, −13.40645607451923673112907701802, −12.843725003841024875448533113, −12.33952706221279248063502359472, −11.127717027480714942339983708130, −11.05018748191704405976770296604, −10.13620629710641941301954310372, −9.21759782569217840075975045522, −8.4620378418992972145772274789, −8.00155243348164865338460731000, −7.116069788791635088880240413468, −6.49097629144540824160684925913, −5.60173812751802012808902035714, −4.72153916040570998501949598287, −3.83751968373278968519815200634, −3.60172651178319755891620564400, −2.49820553866813793559335916766, −1.19277900687916826488716647214, −0.58959760368188677386263425010, 0.735143447532392143062359056261, 2.08033938006337553711037756443, 2.73984548121600198367117200873, 3.439309656807625851197315497441, 4.45720459217454234117859522400, 4.94975960026758600707689773236, 6.02080887240038347999551737213, 6.75904893748230610800779770186, 7.30224575024731527650693343278, 8.17282316206536067281265577043, 8.818137319031634810286207275322, 9.61119058119661183540804120585, 10.281138884477087609351774846346, 11.12198861456622772026860192027, 11.884654915934859420349034975044, 12.24189815802084170659228279915, 12.97743539326078700474230973005, 13.87698844618170610103214443274, 14.76010921191674011388079178298, 15.2688111630163294251581129401, 15.61235543553151938305982163776, 16.499428674980998766473027442035, 17.22877326267513593111448728930, 18.03100092888489961774866718624, 18.74537403499474058281056262731

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