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Properties

Label 1-1441-1441.427-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.475 - 0.879i$
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  + (0.926 − 0.377i)2-s + (0.485 + 0.873i)3-s + (0.715 − 0.698i)4-s + (0.981 − 0.192i)5-s + (0.779 + 0.626i)6-s + (−0.354 − 0.935i)7-s + (0.399 − 0.916i)8-s + (−0.527 + 0.849i)9-s + (0.836 − 0.548i)10-s + (0.958 + 0.285i)12-s + (−0.354 − 0.935i)13-s + (−0.681 − 0.732i)14-s + (0.644 + 0.764i)15-s + (0.0241 − 0.999i)16-s + (0.958 − 0.285i)17-s + (−0.168 + 0.985i)18-s + ⋯
L(s)  = 1  + (0.926 − 0.377i)2-s + (0.485 + 0.873i)3-s + (0.715 − 0.698i)4-s + (0.981 − 0.192i)5-s + (0.779 + 0.626i)6-s + (−0.354 − 0.935i)7-s + (0.399 − 0.916i)8-s + (−0.527 + 0.849i)9-s + (0.836 − 0.548i)10-s + (0.958 + 0.285i)12-s + (−0.354 − 0.935i)13-s + (−0.681 − 0.732i)14-s + (0.644 + 0.764i)15-s + (0.0241 − 0.999i)16-s + (0.958 − 0.285i)17-s + (−0.168 + 0.985i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.475 - 0.879i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.475 - 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.205634194 - 1.910759049i\)
\(L(\frac12)\) \(\approx\) \(3.205634194 - 1.910759049i\)
\(L(1)\) \(\approx\) \(2.243750386 - 0.5827362408i\)
\(L(1)\) \(\approx\) \(2.243750386 - 0.5827362408i\)

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 \( 1 + (0.485 + 0.873i)T \)
5 \( 1 + (0.981 - 0.192i)T \)
7 \( 1 + (-0.354 - 0.935i)T \)
13 \( 1 + (-0.354 - 0.935i)T \)
17 \( 1 + (0.958 - 0.285i)T \)
19 \( 1 + (-0.607 - 0.794i)T \)
23 \( 1 + (0.215 - 0.976i)T \)
29 \( 1 + (-0.970 + 0.239i)T \)
31 \( 1 + (-0.861 + 0.506i)T \)
37 \( 1 + (0.885 + 0.464i)T \)
41 \( 1 + (-0.943 + 0.331i)T \)
43 \( 1 + (0.981 - 0.192i)T \)
47 \( 1 + (-0.527 + 0.849i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.958 - 0.285i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.981 + 0.192i)T \)
71 \( 1 + (0.995 + 0.0965i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.681 + 0.732i)T \)
83 \( 1 + (0.715 + 0.698i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
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

−21.1396467602959773239797383463, −20.27611732208461158299736931373, −19.17037978678483785243435164932, −18.75353072486216094044586264806, −17.82452045258841257695545708204, −16.98310532820809709105527947251, −16.403630095524401659772451475685, −15.1488868522916877914921024917, −14.61957906845708846791362526826, −14.11423652832315576113225724131, −13.13254987998426448350123187389, −12.78936139947956058471402362040, −11.94973416636946977656497097670, −11.25112518159190879007414178921, −9.85034929826444018229041450722, −9.14880520150193343229938237892, −8.2390649619378959401040054749, −7.333346146554106301341940409258, −6.602876160029826188962678647054, −5.79683558851232395619981625310, −5.4413659152051024039654873233, −3.9069449169624088785296525792, −3.10783087086130303089306624061, −2.09954058734996083921290655650, −1.77131854416287786097192965541, 0.90456462639187929896756223496, 2.19044896301853989931483591529, 2.97558088944784693400268739286, 3.69362687770744794105851616231, 4.73938044968220620171791866921, 5.22549429384255594601818218644, 6.17523251781562426614341109286, 7.11045806693699251466034951096, 8.135541854877171540260152609645, 9.435469414392528012272429551141, 9.82619208634426932610066334140, 10.680718454453390960836912100609, 11.042333782151882720605212780229, 12.58753203494203576258441910426, 13.002163282779113631897086608790, 13.82549650020011549459380989832, 14.452769370207260102727171665048, 14.99675182219720758903408690453, 16.02283701564789876207394777477, 16.67781692111668685909825388388, 17.28445508523735865515791333622, 18.54891914723334751960439823447, 19.46445001420710854357682687776, 20.34544156482627797222328486174, 20.48880386279918686564353054387

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