Properties

Label 1-1441-1441.292-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.999 + 0.0156i$
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  + (0.399 − 0.916i)2-s + (0.779 + 0.626i)3-s + (−0.681 − 0.732i)4-s + (−0.262 − 0.964i)5-s + (0.885 − 0.464i)6-s + (0.443 + 0.896i)7-s + (−0.943 + 0.331i)8-s + (0.215 + 0.976i)9-s + (−0.989 − 0.144i)10-s + (−0.0724 − 0.997i)12-s + (−0.715 + 0.698i)13-s + (0.998 − 0.0483i)14-s + (0.399 − 0.916i)15-s + (−0.0724 + 0.997i)16-s + (−0.970 + 0.239i)17-s + (0.981 + 0.192i)18-s + ⋯
L(s)  = 1  + (0.399 − 0.916i)2-s + (0.779 + 0.626i)3-s + (−0.681 − 0.732i)4-s + (−0.262 − 0.964i)5-s + (0.885 − 0.464i)6-s + (0.443 + 0.896i)7-s + (−0.943 + 0.331i)8-s + (0.215 + 0.976i)9-s + (−0.989 − 0.144i)10-s + (−0.0724 − 0.997i)12-s + (−0.715 + 0.698i)13-s + (0.998 − 0.0483i)14-s + (0.399 − 0.916i)15-s + (−0.0724 + 0.997i)16-s + (−0.970 + 0.239i)17-s + (0.981 + 0.192i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.999 + 0.0156i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.999 + 0.0156i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.031616272 + 0.01594781592i\)
\(L(\frac12)\) \(\approx\) \(2.031616272 + 0.01594781592i\)
\(L(1)\) \(\approx\) \(1.406745970 - 0.3159339882i\)
\(L(1)\) \(\approx\) \(1.406745970 - 0.3159339882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.399 + 0.916i)T \)
3 \( 1 + (-0.779 - 0.626i)T \)
5 \( 1 + (0.262 + 0.964i)T \)
7 \( 1 + (-0.443 - 0.896i)T \)
13 \( 1 + (0.715 - 0.698i)T \)
17 \( 1 + (0.970 - 0.239i)T \)
19 \( 1 + (0.0724 + 0.997i)T \)
23 \( 1 + (-0.943 - 0.331i)T \)
29 \( 1 + (-0.995 + 0.0965i)T \)
31 \( 1 + (-0.943 + 0.331i)T \)
37 \( 1 + (0.485 - 0.873i)T \)
41 \( 1 + (-0.970 - 0.239i)T \)
43 \( 1 + (-0.998 - 0.0483i)T \)
47 \( 1 + (0.399 - 0.916i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.644 + 0.764i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.998 + 0.0483i)T \)
71 \( 1 + (-0.943 + 0.331i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.262 + 0.964i)T \)
83 \( 1 + (-0.485 - 0.873i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.443 - 0.896i)T \)
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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

−20.84442735342134130121652697374, −19.80255062317914672527647648439, −19.25387548914430717403566703473, −18.23888363980913301912891174280, −17.74495317203603134684383371202, −17.1192567577395902254338213594, −15.92445128496084117254487947243, −15.25281459885829558042339155991, −14.44133707848499329082918478304, −14.20517450140874566654603782554, −13.34819604831854906783520560079, −12.593937131537839250939702650967, −11.75866860553031104765451176318, −10.63331838961773082549814300848, −9.83443433760367447711904999610, −8.675182601812328830955481974773, −8.02630870030841811943984774627, −7.242120120251593453343738090467, −6.92211063247938751139431228221, −5.98961287298911436625088140039, −4.73665166265091707601783971944, −3.917070969752144973778212541663, −3.103241496745822771463134019534, −2.27965569301992821542470525694, −0.657136848027008977314020711133, 1.19001382173716120828893780172, 2.28580619481524059394206332914, 2.77729885526893457592784807936, 4.0497691623930831752808066299, 4.782351791057656683589445135730, 5.02616563384312328160976915056, 6.36228612347430948873721955374, 7.80921331256892998288613837862, 8.72791006670671755005708936974, 9.10523586083902219845251478198, 9.7287648493360635014878500598, 10.89467331550982996844643756431, 11.524365294901770898140949949119, 12.325947246288187428005299434437, 13.08621616096423854580223782265, 13.79285532669806267216501358719, 14.60023324890242954607051606661, 15.452379582221444976124698676240, 15.74258208150353747720061267664, 17.10096161381645343261208546991, 17.72121647277329164942311380503, 19.02953196398965892836271148830, 19.34818922352517218963045205434, 20.049359914342053874008753563227, 20.82672717142586407583780161038

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