Properties

Label 1-1441-1441.219-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.636 - 0.771i$
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  + (0.958 − 0.285i)2-s + (−0.443 − 0.896i)3-s + (0.836 − 0.548i)4-s + (0.715 − 0.698i)5-s + (−0.681 − 0.732i)6-s + (0.906 + 0.421i)7-s + (0.644 − 0.764i)8-s + (−0.607 + 0.794i)9-s + (0.485 − 0.873i)10-s + (−0.861 − 0.506i)12-s + (−0.485 − 0.873i)13-s + (0.989 + 0.144i)14-s + (−0.943 − 0.331i)15-s + (0.399 − 0.916i)16-s + (0.995 + 0.0965i)17-s + (−0.354 + 0.935i)18-s + ⋯
L(s)  = 1  + (0.958 − 0.285i)2-s + (−0.443 − 0.896i)3-s + (0.836 − 0.548i)4-s + (0.715 − 0.698i)5-s + (−0.681 − 0.732i)6-s + (0.906 + 0.421i)7-s + (0.644 − 0.764i)8-s + (−0.607 + 0.794i)9-s + (0.485 − 0.873i)10-s + (−0.861 − 0.506i)12-s + (−0.485 − 0.873i)13-s + (0.989 + 0.144i)14-s + (−0.943 − 0.331i)15-s + (0.399 − 0.916i)16-s + (0.995 + 0.0965i)17-s + (−0.354 + 0.935i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.636 - 0.771i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.636 - 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.300567154 - 2.759550922i\)
\(L(\frac12)\) \(\approx\) \(1.300567154 - 2.759550922i\)
\(L(1)\) \(\approx\) \(1.548264415 - 1.189247166i\)
\(L(1)\) \(\approx\) \(1.548264415 - 1.189247166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.958 - 0.285i)T \)
3 \( 1 + (-0.443 - 0.896i)T \)
5 \( 1 + (0.715 - 0.698i)T \)
7 \( 1 + (0.906 + 0.421i)T \)
13 \( 1 + (-0.485 - 0.873i)T \)
17 \( 1 + (0.995 + 0.0965i)T \)
19 \( 1 + (-0.748 + 0.663i)T \)
23 \( 1 + (-0.926 + 0.377i)T \)
29 \( 1 + (-0.607 - 0.794i)T \)
31 \( 1 + (0.527 + 0.849i)T \)
37 \( 1 + (-0.779 - 0.626i)T \)
41 \( 1 + (-0.399 - 0.916i)T \)
43 \( 1 + (0.168 - 0.985i)T \)
47 \( 1 + (-0.568 - 0.822i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.215 + 0.976i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.168 + 0.985i)T \)
71 \( 1 + (0.970 - 0.239i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (-0.998 + 0.0483i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.981 + 0.192i)T \)
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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

−21.19371361477620233304548035451, −20.71525589970345888363809343684, −19.80100540289642003422300153070, −18.61220082881255562998409067924, −17.655634595738752448008865284797, −17.01116455288154688912988045435, −16.56130413669818846657053586839, −15.52788905295628372380705527558, −14.66499442864120461408145704865, −14.42879003478590584592733467416, −13.69320857763405494527079654024, −12.607717576108011227965802882427, −11.607907370164215649103086912239, −11.19941400749137877001242902784, −10.35370727340372117654326321852, −9.67539340922235999515552544002, −8.48269163974955783139560580028, −7.46902133490013830636268925946, −6.59098538058764357249206394356, −5.93856126474991556563025697021, −4.99717412075211923491002034533, −4.470414675955825513439025209925, −3.54391647981234423632346949415, −2.58866403934043042412537302937, −1.60211130427966211973320174260, 0.846618896607377672733347968192, 1.90775067764508589881354684600, 2.21967752026166944820529740849, 3.62290636557003429233152960511, 4.82288875741309558487922004584, 5.60385803381538209403780189178, 5.73660062505045665153667694168, 6.948855659484484513345896139123, 7.89108589958066378222648374014, 8.53687950575988002688182803059, 9.99961775758946219493650708943, 10.52493261784203534608744912849, 11.64479210413673987182136772240, 12.23929632763387252535509515169, 12.64687592083680246794491330663, 13.56931221224154697238117096426, 14.15920515773820270488763455515, 14.86571394073613141535305737262, 15.87656233663603091554107132044, 16.83008580305431924281647452955, 17.41113095503050610444990576550, 18.174181790980813892557910193450, 19.05938916788739118883691818455, 19.776908928412566755168722782043, 20.73175547745640003465961929702

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