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Properties

Label 1-1441-1441.81-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.0710 + 0.997i$
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  + (−0.861 + 0.506i)2-s + (0.779 + 0.626i)3-s + (0.485 − 0.873i)4-s + (−0.262 − 0.964i)5-s + (−0.989 − 0.144i)6-s + (0.885 + 0.464i)7-s + (0.0241 + 0.999i)8-s + (0.215 + 0.976i)9-s + (0.715 + 0.698i)10-s + (0.926 − 0.377i)12-s + (0.885 + 0.464i)13-s + (−0.998 + 0.0483i)14-s + (0.399 − 0.916i)15-s + (−0.527 − 0.849i)16-s + (0.926 + 0.377i)17-s + (−0.681 − 0.732i)18-s + ⋯
L(s)  = 1  + (−0.861 + 0.506i)2-s + (0.779 + 0.626i)3-s + (0.485 − 0.873i)4-s + (−0.262 − 0.964i)5-s + (−0.989 − 0.144i)6-s + (0.885 + 0.464i)7-s + (0.0241 + 0.999i)8-s + (0.215 + 0.976i)9-s + (0.715 + 0.698i)10-s + (0.926 − 0.377i)12-s + (0.885 + 0.464i)13-s + (−0.998 + 0.0483i)14-s + (0.399 − 0.916i)15-s + (−0.527 − 0.849i)16-s + (0.926 + 0.377i)17-s + (−0.681 − 0.732i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0710 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0710 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.0710 + 0.997i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.0710 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.172534515 + 1.091996063i\)
\(L(\frac12)\) \(\approx\) \(1.172534515 + 1.091996063i\)
\(L(1)\) \(\approx\) \(0.9909406418 + 0.4412525595i\)
\(L(1)\) \(\approx\) \(0.9909406418 + 0.4412525595i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.861 + 0.506i)T \)
3 \( 1 + (0.779 + 0.626i)T \)
5 \( 1 + (-0.262 - 0.964i)T \)
7 \( 1 + (0.885 + 0.464i)T \)
13 \( 1 + (0.885 + 0.464i)T \)
17 \( 1 + (0.926 + 0.377i)T \)
19 \( 1 + (0.644 + 0.764i)T \)
23 \( 1 + (0.958 - 0.285i)T \)
29 \( 1 + (-0.748 + 0.663i)T \)
31 \( 1 + (-0.943 + 0.331i)T \)
37 \( 1 + (0.120 + 0.992i)T \)
41 \( 1 + (-0.0724 - 0.997i)T \)
43 \( 1 + (-0.262 - 0.964i)T \)
47 \( 1 + (0.215 + 0.976i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.926 + 0.377i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.262 + 0.964i)T \)
71 \( 1 + (-0.607 - 0.794i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.998 - 0.0483i)T \)
83 \( 1 + (0.485 + 0.873i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
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

−20.30643804891240459785623195073, −19.81326931791354504268248396843, −18.91261766825841910670911754928, −18.3479046601283707313372355614, −17.94209196800781261980966326819, −17.0994270029509954538475603846, −16.031977704132425441872170542743, −15.12617567503783972636692464413, −14.5526509672506138899611282253, −13.55311383175415341247839706373, −13.02521838162502925806505445451, −11.7826600009796237456953235839, −11.33363433075774974050437827576, −10.58158862676876975563215474387, −9.65848223403394195813923460097, −8.86937202132303588444348372483, −7.92186421078074861715637762332, −7.49955067506520328520079379426, −6.91537653129710558267444141681, −5.7391240122876891690885595153, −4.120492073252661019401750301218, −3.33049706792824883671430212514, −2.70416858871779244233245994648, −1.646202087273214633383465218, −0.8170631278399444046509861316, 1.32466247259782766110101521380, 1.78437395765522077814910603681, 3.214160296501390850010754963968, 4.2375742406188486788935162263, 5.25546762545307134897895782289, 5.65320648710477090673286319802, 7.19734803693244446847863467889, 7.91548488603245420531579326216, 8.65241306480576019038340543896, 8.94352335530451683567315976310, 9.8424730190768420315655527163, 10.752321169695489534989535919155, 11.47572123310761208507277622784, 12.42135529784361455516323076218, 13.57260903943053332170352681856, 14.37779834716238876451292162772, 14.9744676857730739899735301602, 15.70434364763990998619728166061, 16.49002003702983313073671565285, 16.80808365866910860645707423986, 17.92963741560663577891666890237, 18.796924520632921071275556673132, 19.20500699018666991115565444938, 20.39961749822910622360918589290, 20.64895191998698941614494610780

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