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Properties

Label 1-1441-1441.780-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.964 + 0.263i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (−0.926 − 0.377i)2-s + (−0.906 + 0.421i)3-s + (0.715 + 0.698i)4-s + (0.485 − 0.873i)5-s + (0.998 − 0.0483i)6-s + (−0.836 + 0.548i)7-s + (−0.399 − 0.916i)8-s + (0.644 − 0.764i)9-s + (−0.779 + 0.626i)10-s + (−0.943 − 0.331i)12-s + (−0.779 − 0.626i)13-s + (0.981 − 0.192i)14-s + (−0.0724 + 0.997i)15-s + (0.0241 + 0.999i)16-s + (0.607 + 0.794i)17-s + (−0.885 + 0.464i)18-s + ⋯
L(s)  = 1  + (−0.926 − 0.377i)2-s + (−0.906 + 0.421i)3-s + (0.715 + 0.698i)4-s + (0.485 − 0.873i)5-s + (0.998 − 0.0483i)6-s + (−0.836 + 0.548i)7-s + (−0.399 − 0.916i)8-s + (0.644 − 0.764i)9-s + (−0.779 + 0.626i)10-s + (−0.943 − 0.331i)12-s + (−0.779 − 0.626i)13-s + (0.981 − 0.192i)14-s + (−0.0724 + 0.997i)15-s + (0.0241 + 0.999i)16-s + (0.607 + 0.794i)17-s + (−0.885 + 0.464i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.964 + 0.263i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (780, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.964 + 0.263i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3309372896 + 0.04442175709i\)
\(L(\frac12)\) \(\approx\) \(0.3309372896 + 0.04442175709i\)
\(L(1)\) \(\approx\) \(0.4267042200 - 0.07238746126i\)
\(L(1)\) \(\approx\) \(0.4267042200 - 0.07238746126i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.926 - 0.377i)T \)
3 \( 1 + (-0.906 + 0.421i)T \)
5 \( 1 + (0.485 - 0.873i)T \)
7 \( 1 + (-0.836 + 0.548i)T \)
13 \( 1 + (-0.779 - 0.626i)T \)
17 \( 1 + (0.607 + 0.794i)T \)
19 \( 1 + (-0.568 - 0.822i)T \)
23 \( 1 + (-0.861 + 0.506i)T \)
29 \( 1 + (-0.644 - 0.764i)T \)
31 \( 1 + (0.215 - 0.976i)T \)
37 \( 1 + (-0.989 - 0.144i)T \)
41 \( 1 + (-0.0241 + 0.999i)T \)
43 \( 1 + (0.681 + 0.732i)T \)
47 \( 1 + (-0.970 + 0.239i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.958 + 0.285i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.681 + 0.732i)T \)
71 \( 1 + (-0.748 + 0.663i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.120 - 0.992i)T \)
83 \( 1 + (0.443 + 0.896i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.262 - 0.964i)T \)
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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

−20.32788434429744173949289906932, −19.216229450787628501272886684040, −18.974102687772294666974676364633, −18.21886998612361870659467368406, −17.488474702592564990952550927016, −16.81295642254477037689265476327, −16.310533180574441049565931844516, −15.51222400217897505222503833082, −14.23012559825782235071947230698, −14.03730349176853166039283286051, −12.64231579195293026722535469873, −11.98491546713994530342975612492, −11.04619824459196872032445261964, −10.21819628156051032922530971026, −10.0347344617861027866043614927, −8.97783195209432667378995553381, −7.6458505541302967495251368622, −7.11348698813120185133186005804, −6.52996741078159538797593857997, −5.855020746599567383370774610694, −4.93894224448656753987551147613, −3.488311742735742401387175768754, −2.307036479479757475716777209847, −1.52274448743198539333046088605, −0.22266074146326481354933504869, 0.41167633892807445890743588172, 1.536420460564543667424728242840, 2.560741267087172802462004138579, 3.67789431121562937654375700564, 4.66545570157121888503807773454, 5.83517278332822407792291925827, 6.152139018478205042699303210695, 7.346979876281879408322472885361, 8.310433026197395467798024485187, 9.21089475177344436976791075428, 9.87563387181660650946914105483, 10.19694832163399704659786682671, 11.386335584414806530899148567295, 12.02746086707384918391944275606, 12.797851298725572543673865210693, 13.13539402813008972121890364124, 14.91632303521094491180758987646, 15.583742927218779535932949994615, 16.34071397263500684151811252018, 16.87056274436411466411360830624, 17.57578389281190630701976212095, 18.019881551688492401777711289891, 19.26049861883026732523005370262, 19.55720679156231671059377972629, 20.736167004980445410419474598526

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