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Properties

Label 1-1441-1441.1061-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.497 + 0.867i$
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  + (−0.970 − 0.239i)2-s + (0.485 + 0.873i)3-s + (0.885 + 0.464i)4-s + (0.981 − 0.192i)5-s + (−0.262 − 0.964i)6-s + (0.836 + 0.548i)7-s + (−0.748 − 0.663i)8-s + (−0.527 + 0.849i)9-s + (−0.998 − 0.0483i)10-s + (0.0241 + 0.999i)12-s + (0.779 − 0.626i)13-s + (−0.681 − 0.732i)14-s + (0.644 + 0.764i)15-s + (0.568 + 0.822i)16-s + (−0.943 − 0.331i)17-s + (0.715 − 0.698i)18-s + ⋯
L(s)  = 1  + (−0.970 − 0.239i)2-s + (0.485 + 0.873i)3-s + (0.885 + 0.464i)4-s + (0.981 − 0.192i)5-s + (−0.262 − 0.964i)6-s + (0.836 + 0.548i)7-s + (−0.748 − 0.663i)8-s + (−0.527 + 0.849i)9-s + (−0.998 − 0.0483i)10-s + (0.0241 + 0.999i)12-s + (0.779 − 0.626i)13-s + (−0.681 − 0.732i)14-s + (0.644 + 0.764i)15-s + (0.568 + 0.822i)16-s + (−0.943 − 0.331i)17-s + (0.715 − 0.698i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.497 + 0.867i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1061, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.497 + 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.464743317 + 0.8484815178i\)
\(L(\frac12)\) \(\approx\) \(1.464743317 + 0.8484815178i\)
\(L(1)\) \(\approx\) \(1.077487939 + 0.3000381860i\)
\(L(1)\) \(\approx\) \(1.077487939 + 0.3000381860i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.970 - 0.239i)T \)
3 \( 1 + (0.485 + 0.873i)T \)
5 \( 1 + (0.981 - 0.192i)T \)
7 \( 1 + (0.836 + 0.548i)T \)
13 \( 1 + (0.779 - 0.626i)T \)
17 \( 1 + (-0.943 - 0.331i)T \)
19 \( 1 + (0.958 + 0.285i)T \)
23 \( 1 + (0.399 + 0.916i)T \)
29 \( 1 + (0.644 - 0.764i)T \)
31 \( 1 + (-0.861 + 0.506i)T \)
37 \( 1 + (-0.989 + 0.144i)T \)
41 \( 1 + (-0.607 - 0.794i)T \)
43 \( 1 + (0.485 + 0.873i)T \)
47 \( 1 + (-0.0724 - 0.997i)T \)
53 \( 1 + T \)
59 \( 1 + (0.568 + 0.822i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.485 - 0.873i)T \)
71 \( 1 + (0.215 + 0.976i)T \)
73 \( 1 + T \)
79 \( 1 + (0.485 + 0.873i)T \)
83 \( 1 + (0.715 + 0.698i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.998 + 0.0483i)T \)
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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

−20.4891821756717250178299767355, −19.824279778621740976885107535668, −18.82393654546424799520862323816, −18.255431369001412018296587055328, −17.75845303127533867996292050268, −17.14580559200949330890456034962, −16.32010825542141686313099538338, −15.23173953285673873243711138054, −14.40282772171659376486697467713, −13.925030738640953000263058684682, −13.161100770082549648789990115, −12.07934581549128839147038968857, −11.10745547903847259784783008808, −10.657793343447640649150381856413, −9.509670334097559213267513165455, −8.83701373173316900388629853433, −8.277995117096764888695516486173, −7.19276594860031517115334433702, −6.76245132935348245487711153058, −5.96083991166981132021657988289, −4.9241711192994269055231478820, −3.42505330427756158641729139237, −2.30743274316983777343202472138, −1.69533264261080037512709886913, −0.91918411655401292503653655207, 1.227265316319283163488047987561, 2.11156392577362906566654021741, 2.87260571104165920555117542599, 3.85261102916569162236117649201, 5.19322372720741537010809091516, 5.6564441782073786457948900640, 6.89891432802583806754457160390, 7.9818589335430630924883008858, 8.66913220360335149397001216246, 9.18325120294163045658408885099, 9.93322734413904051456421663340, 10.70838032111061443629477705918, 11.29990733974486963801113295686, 12.22527337039995871075595276341, 13.42094262848346590213706850176, 13.97513141362647829952458632196, 15.1322192170509545125437982843, 15.57017344138177525858104637470, 16.3830507378896269808585875934, 17.222904063881111917231704223, 17.939134464612282891616156030940, 18.32408784859858087611150116327, 19.49366818669836826507465533325, 20.21617292022610809190079223393, 20.860327799792158991796823815002

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