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Properties

Label 1-1441-1441.1125-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.0104 - 0.999i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (0.981 − 0.192i)2-s + (−0.861 − 0.506i)3-s + (0.926 − 0.377i)4-s + (0.995 − 0.0965i)5-s + (−0.943 − 0.331i)6-s + (0.568 − 0.822i)7-s + (0.836 − 0.548i)8-s + (0.485 + 0.873i)9-s + (0.958 − 0.285i)10-s + (−0.989 − 0.144i)12-s + (0.568 − 0.822i)13-s + (0.399 − 0.916i)14-s + (−0.906 − 0.421i)15-s + (0.715 − 0.698i)16-s + (−0.989 + 0.144i)17-s + (0.644 + 0.764i)18-s + ⋯
L(s)  = 1  + (0.981 − 0.192i)2-s + (−0.861 − 0.506i)3-s + (0.926 − 0.377i)4-s + (0.995 − 0.0965i)5-s + (−0.943 − 0.331i)6-s + (0.568 − 0.822i)7-s + (0.836 − 0.548i)8-s + (0.485 + 0.873i)9-s + (0.958 − 0.285i)10-s + (−0.989 − 0.144i)12-s + (0.568 − 0.822i)13-s + (0.399 − 0.916i)14-s + (−0.906 − 0.421i)15-s + (0.715 − 0.698i)16-s + (−0.989 + 0.144i)17-s + (0.644 + 0.764i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0104 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0104 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.0104 - 0.999i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.0104 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.167617942 - 2.190363483i\)
\(L(\frac12)\) \(\approx\) \(2.167617942 - 2.190363483i\)
\(L(1)\) \(\approx\) \(1.736934892 - 0.8522809287i\)
\(L(1)\) \(\approx\) \(1.736934892 - 0.8522809287i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.981 - 0.192i)T \)
3 \( 1 + (-0.861 - 0.506i)T \)
5 \( 1 + (0.995 - 0.0965i)T \)
7 \( 1 + (0.568 - 0.822i)T \)
13 \( 1 + (0.568 - 0.822i)T \)
17 \( 1 + (-0.989 + 0.144i)T \)
19 \( 1 + (-0.443 + 0.896i)T \)
23 \( 1 + (0.779 - 0.626i)T \)
29 \( 1 + (0.120 + 0.992i)T \)
31 \( 1 + (-0.262 - 0.964i)T \)
37 \( 1 + (-0.970 - 0.239i)T \)
41 \( 1 + (-0.168 - 0.985i)T \)
43 \( 1 + (0.995 - 0.0965i)T \)
47 \( 1 + (0.485 + 0.873i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.989 + 0.144i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.995 + 0.0965i)T \)
71 \( 1 + (-0.998 - 0.0483i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.399 + 0.916i)T \)
83 \( 1 + (0.926 + 0.377i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.0241 + 0.999i)T \)
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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

−21.10488042095895309861693564779, −20.7952096725833672070958528877, −19.50849403681020597735215939192, −18.43488460854202243778061401263, −17.60855504486046068302742064918, −17.19646375635236593259154711104, −16.252788596772983604448427372855, −15.48691689034212532469434892840, −15.008078036699642399425439423691, −14.02748890057174242602154589628, −13.34424671395153037154840204713, −12.56388315493432539012235325104, −11.61780449728177923775743166309, −11.17428215495819563344108419289, −10.43813633779645155490089586382, −9.251413977345000711504411978509, −8.68735343558811355878917393175, −7.14688434980126020069781863275, −6.43469495966658856059548739966, −5.88672114724808568399647342627, −4.946402820272804531789118748298, −4.60967657444203248439180129638, −3.35879895831665964868521499559, −2.284577737022940565332557809315, −1.49374030267713984817408474844, 0.97256952191490440203892175650, 1.695769570292602911239922644135, 2.593849119540270804712844264794, 3.93972579393965365125449961232, 4.73711154744583211182057575681, 5.53904132377794765740148840596, 6.14082170280860762581586643874, 6.92820331194904145412962664620, 7.68564317251309516733776612624, 8.878372581180680754998843375152, 10.396408746597010561380412951099, 10.58776155286179130704319850591, 11.23792297173583812611003037890, 12.460720769411788509948471056224, 12.832561979755142399322250438272, 13.6404992525258829660689555631, 14.12735769632771322870563643480, 15.10281199754062265421024874763, 16.05316193877739225372941784914, 16.91711796670653929535651432573, 17.351316824294226287845715614601, 18.21729545245037110605106964591, 18.985435144243743811640909489170, 20.09823618646909391436635605932, 20.708433700707493343647016029045

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