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Properties

Label 1-1441-1441.294-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.0853 + 0.996i$
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  + (−0.527 + 0.849i)2-s + (0.485 + 0.873i)3-s + (−0.443 − 0.896i)4-s + (0.981 − 0.192i)5-s + (−0.998 − 0.0483i)6-s + (0.262 − 0.964i)7-s + (0.995 + 0.0965i)8-s + (−0.527 + 0.849i)9-s + (−0.354 + 0.935i)10-s + (0.568 − 0.822i)12-s + (0.998 − 0.0483i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (−0.607 + 0.794i)16-s + (−0.607 + 0.794i)17-s + (−0.443 − 0.896i)18-s + ⋯
L(s)  = 1  + (−0.527 + 0.849i)2-s + (0.485 + 0.873i)3-s + (−0.443 − 0.896i)4-s + (0.981 − 0.192i)5-s + (−0.998 − 0.0483i)6-s + (0.262 − 0.964i)7-s + (0.995 + 0.0965i)8-s + (−0.527 + 0.849i)9-s + (−0.354 + 0.935i)10-s + (0.568 − 0.822i)12-s + (0.998 − 0.0483i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (−0.607 + 0.794i)16-s + (−0.607 + 0.794i)17-s + (−0.443 − 0.896i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.0853 + 0.996i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (294, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.0853 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.343296335 + 1.233089532i\)
\(L(\frac12)\) \(\approx\) \(1.343296335 + 1.233089532i\)
\(L(1)\) \(\approx\) \(1.038340894 + 0.6132607066i\)
\(L(1)\) \(\approx\) \(1.038340894 + 0.6132607066i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.527 + 0.849i)T \)
3 \( 1 + (0.485 + 0.873i)T \)
5 \( 1 + (0.981 - 0.192i)T \)
7 \( 1 + (0.262 - 0.964i)T \)
13 \( 1 + (0.998 - 0.0483i)T \)
17 \( 1 + (-0.607 + 0.794i)T \)
19 \( 1 + (0.0241 + 0.999i)T \)
23 \( 1 + (0.748 - 0.663i)T \)
29 \( 1 + (0.926 + 0.377i)T \)
31 \( 1 + (0.861 - 0.506i)T \)
37 \( 1 + (0.443 + 0.896i)T \)
41 \( 1 + (-0.0241 - 0.999i)T \)
43 \( 1 + (-0.120 + 0.992i)T \)
47 \( 1 + (-0.926 + 0.377i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.0241 - 0.999i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.120 - 0.992i)T \)
71 \( 1 + (-0.399 + 0.916i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.906 - 0.421i)T \)
83 \( 1 + (0.715 + 0.698i)T \)
89 \( 1 + T \)
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

−20.514788976570074546927081289195, −19.68959937689649987547161176133, −18.98853835314051067574682538090, −18.23531663149621779564453236411, −17.88229636263747789485246864351, −17.3325150194391612247025804534, −16.09442479561711024655784023215, −15.16737381493423941903261864224, −14.13751037853413612504658656545, −13.348142528851219828347502019247, −13.15654223941821713334234139184, −11.96439486411194483331085765219, −11.48638932930495602054558270947, −10.57780853333855954448979895822, −9.470099431606923067885661129481, −8.93048937481493367168675628691, −8.46052762949430128804300257805, −7.30862703595852847845776428822, −6.54587433273933820827647913391, −5.589950430456893860842230924001, −4.51281818579861047340256875318, −2.993270162907262770544191348142, −2.69975797968671691817143699519, −1.76153316719197610818592444053, −0.97501533014884657766279885407, 1.07886548747070355512968401116, 1.97958663032063239346613643094, 3.385927683651753319609839281415, 4.42962405084891486648106590998, 4.97680403031952026682335212559, 6.10852083324101566722317515029, 6.60627829383231383726258416087, 7.976139498116193208118593277373, 8.418357622751744573879375677770, 9.21941370305396354695257310605, 10.11394343561935347472249515857, 10.46223757585945088438468878391, 11.22016168717314602137425800807, 13.01719032534992841231392356760, 13.54070879628603572530642567037, 14.30567745190104749963486325198, 14.76104772776971709212295855638, 15.78380127053872863301654070421, 16.41613710455560303231898924589, 17.07960549248424372810179504863, 17.59252411772482097194357142859, 18.5331178235501976575711082396, 19.37620689247329964239744189894, 20.28252073821058340966016773827, 20.78288895672050095594759301619

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