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Properties

Label 1-4056-4056.2885-r1-0-0
Degree $1$
Conductor $4056$
Sign $-0.230 - 0.973i$
Analytic cond. $435.877$
Root an. cond. $435.877$
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  + (−0.120 + 0.992i)5-s + (−0.885 + 0.464i)7-s + (0.354 + 0.935i)11-s + (−0.885 + 0.464i)17-s + 19-s − 23-s + (−0.970 − 0.239i)25-s + (−0.354 + 0.935i)29-s + (0.970 − 0.239i)31-s + (−0.354 − 0.935i)35-s + (−0.970 + 0.239i)37-s + (0.568 + 0.822i)41-s + (0.970 + 0.239i)43-s + (−0.748 + 0.663i)47-s + (0.568 − 0.822i)49-s + ⋯
L(s)  = 1  + (−0.120 + 0.992i)5-s + (−0.885 + 0.464i)7-s + (0.354 + 0.935i)11-s + (−0.885 + 0.464i)17-s + 19-s − 23-s + (−0.970 − 0.239i)25-s + (−0.354 + 0.935i)29-s + (0.970 − 0.239i)31-s + (−0.354 − 0.935i)35-s + (−0.970 + 0.239i)37-s + (0.568 + 0.822i)41-s + (0.970 + 0.239i)43-s + (−0.748 + 0.663i)47-s + (0.568 − 0.822i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4056\)    =    \(2^{3} \cdot 3 \cdot 13^{2}\)
Sign: $-0.230 - 0.973i$
Analytic conductor: \(435.877\)
Root analytic conductor: \(435.877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4056} (2885, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4056,\ (1:\ ),\ -0.230 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4809387715 + 0.6080311011i\)
\(L(\frac12)\) \(\approx\) \(-0.4809387715 + 0.6080311011i\)
\(L(1)\) \(\approx\) \(0.7296817486 + 0.4000679818i\)
\(L(1)\) \(\approx\) \(0.7296817486 + 0.4000679818i\)

Euler product

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

−17.61923321374752414134190653777, −17.108064732775582169086485515018, −16.26188605971707675996851843378, −15.98847937455477745767301586866, −15.37664775291390752400880907191, −14.07156941275677106637642691179, −13.64755485469861523656407148673, −13.21161691372899835822801619593, −12.13731242294151685171938725382, −11.88981801787114868997902074815, −10.90287788268167616583491794637, −10.13742489717022561288752351902, −9.29779881943950639373191742774, −8.96447510734629573053775872916, −8.02655861004383121402753069852, −7.37641854393224132956251221794, −6.415617050840984188585102670283, −5.85416198658424566524970942824, −5.007101887033330416750686886899, −4.12445922369223821190602221431, −3.59383656953433153805243193657, −2.65319933952043049405001802298, −1.59371286675005093121123381435, −0.58170406297508871281357061800, −0.17583045892187268498278840892, 1.296752276145763289929176204164, 2.29661778411425386819097912331, 2.87800452297914211906189553922, 3.73578003655034468951636290469, 4.40120645993695744600432355055, 5.51093284656616889705989567979, 6.25524325782718028209990698824, 6.822470677624889285493038409579, 7.43060915176469066473807675015, 8.322252452040232504226302303652, 9.247815880888224650931305237700, 9.8323287949869959234302018058, 10.38083226322023295975414620748, 11.276541420583533205240596270013, 11.91960080002178092645650366640, 12.562643375143671781515625856591, 13.31533324846188839085659331990, 14.11200083762096312603826107395, 14.69798683310298743674327024017, 15.53744399423468002410563676546, 15.77493780210043652668107981353, 16.704784072322671077970248296471, 17.66809664628013753328273539636, 18.04935704219361048754311920188, 18.70648644175140659511262751152

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