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Properties

Label 1-1441-1441.26-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.445 - 0.895i$
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  + (−0.568 + 0.822i)2-s + (0.715 + 0.698i)3-s + (−0.354 − 0.935i)4-s + (−0.989 − 0.144i)5-s + (−0.981 + 0.192i)6-s + (−0.906 + 0.421i)7-s + (0.970 + 0.239i)8-s + (0.0241 + 0.999i)9-s + (0.681 − 0.732i)10-s + (0.399 − 0.916i)12-s + (0.485 − 0.873i)13-s + (0.168 − 0.985i)14-s + (−0.607 − 0.794i)15-s + (−0.748 + 0.663i)16-s + (0.861 + 0.506i)17-s + (−0.836 − 0.548i)18-s + ⋯
L(s)  = 1  + (−0.568 + 0.822i)2-s + (0.715 + 0.698i)3-s + (−0.354 − 0.935i)4-s + (−0.989 − 0.144i)5-s + (−0.981 + 0.192i)6-s + (−0.906 + 0.421i)7-s + (0.970 + 0.239i)8-s + (0.0241 + 0.999i)9-s + (0.681 − 0.732i)10-s + (0.399 − 0.916i)12-s + (0.485 − 0.873i)13-s + (0.168 − 0.985i)14-s + (−0.607 − 0.794i)15-s + (−0.748 + 0.663i)16-s + (0.861 + 0.506i)17-s + (−0.836 − 0.548i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.445 - 0.895i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.445 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2465166239 + 0.3978648608i\)
\(L(\frac12)\) \(\approx\) \(-0.2465166239 + 0.3978648608i\)
\(L(1)\) \(\approx\) \(0.5675205195 + 0.4269334675i\)
\(L(1)\) \(\approx\) \(0.5675205195 + 0.4269334675i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.568 + 0.822i)T \)
3 \( 1 + (0.715 + 0.698i)T \)
5 \( 1 + (-0.989 - 0.144i)T \)
7 \( 1 + (-0.906 + 0.421i)T \)
13 \( 1 + (0.485 - 0.873i)T \)
17 \( 1 + (0.861 + 0.506i)T \)
19 \( 1 + (-0.215 - 0.976i)T \)
23 \( 1 + (-0.644 + 0.764i)T \)
29 \( 1 + (0.607 - 0.794i)T \)
31 \( 1 + (-0.926 + 0.377i)T \)
37 \( 1 + (-0.779 + 0.626i)T \)
41 \( 1 + (0.995 + 0.0965i)T \)
43 \( 1 + (0.715 + 0.698i)T \)
47 \( 1 + (0.943 - 0.331i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.748 + 0.663i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.715 + 0.698i)T \)
71 \( 1 + (0.527 - 0.849i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.715 - 0.698i)T \)
83 \( 1 + (-0.836 + 0.548i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.681 + 0.732i)T \)
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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

−19.93202906334353414514463821494, −19.24161841724850569746267382931, −18.68343243859008416657398064979, −18.33227398393092659837381564058, −17.04850191714342974010334245469, −16.299134235466524969167652252559, −15.75107980746393728486963871295, −14.20400449369701932831885440998, −14.11257193313918792196863578509, −12.73465995776745207521122806594, −12.47533522960167672770884747422, −11.71654105291003863218150020913, −10.74796960449116547051759119157, −9.961664042106476613270474412346, −9.03119099739068378738882034492, −8.4924877266134725921326035366, −7.45972110960270759941776008376, −7.170973238135352380718422376816, −6.05954815431508797005308818818, −4.26285018341688782914099074290, −3.70557345662178574694849361406, −3.05573420663943503205763120519, −2.0414541958810531437820501390, −0.965833981070285726535564660778, −0.13157527547013447508626061, 1.046353825749045110876598785886, 2.58323556381791719025074790986, 3.521447112143195258375487282000, 4.27111411582347274191690486097, 5.33620668036965079690834843482, 6.05777635235480549198094048105, 7.285537864361611811615947059755, 7.84177913237440521948809563738, 8.6895331810720068308127311908, 9.14845033999747104173626257012, 10.15734747646170402598957865113, 10.64885981752703318860161141929, 11.76900773684761472463374178280, 12.88992764245405294961437745143, 13.57630380251708375315323322975, 14.60248562038564659509457869309, 15.28469584097495871346602580321, 15.74386439526308135452212451483, 16.2186874170168908634355966438, 17.05675274854253032498655530200, 18.07785903152514188441346694904, 18.9715683316205923838066362305, 19.58210905198828456077693334445, 19.87364017004839442242094723832, 20.90333306047555913449387927155

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