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Properties

Label 1-1441-1441.626-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.855 + 0.517i$
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  + (−0.779 + 0.626i)2-s + (0.0241 + 0.999i)3-s + (0.215 − 0.976i)4-s + (0.958 + 0.285i)5-s + (−0.644 − 0.764i)6-s + (0.527 + 0.849i)7-s + (0.443 + 0.896i)8-s + (−0.998 + 0.0483i)9-s + (−0.926 + 0.377i)10-s + (0.981 + 0.192i)12-s + (−0.926 − 0.377i)13-s + (−0.943 − 0.331i)14-s + (−0.262 + 0.964i)15-s + (−0.906 − 0.421i)16-s + (0.681 − 0.732i)17-s + (0.748 − 0.663i)18-s + ⋯
L(s)  = 1  + (−0.779 + 0.626i)2-s + (0.0241 + 0.999i)3-s + (0.215 − 0.976i)4-s + (0.958 + 0.285i)5-s + (−0.644 − 0.764i)6-s + (0.527 + 0.849i)7-s + (0.443 + 0.896i)8-s + (−0.998 + 0.0483i)9-s + (−0.926 + 0.377i)10-s + (0.981 + 0.192i)12-s + (−0.926 − 0.377i)13-s + (−0.943 − 0.331i)14-s + (−0.262 + 0.964i)15-s + (−0.906 − 0.421i)16-s + (0.681 − 0.732i)17-s + (0.748 − 0.663i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.855 + 0.517i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.855 + 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4410159110 + 1.582034838i\)
\(L(\frac12)\) \(\approx\) \(0.4410159110 + 1.582034838i\)
\(L(1)\) \(\approx\) \(0.7053051092 + 0.5693196671i\)
\(L(1)\) \(\approx\) \(0.7053051092 + 0.5693196671i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.779 + 0.626i)T \)
3 \( 1 + (0.0241 + 0.999i)T \)
5 \( 1 + (0.958 + 0.285i)T \)
7 \( 1 + (0.527 + 0.849i)T \)
13 \( 1 + (-0.926 - 0.377i)T \)
17 \( 1 + (0.681 - 0.732i)T \)
19 \( 1 + (-0.120 - 0.992i)T \)
23 \( 1 + (-0.989 - 0.144i)T \)
29 \( 1 + (0.998 + 0.0483i)T \)
31 \( 1 + (0.715 - 0.698i)T \)
37 \( 1 + (-0.861 - 0.506i)T \)
41 \( 1 + (0.906 - 0.421i)T \)
43 \( 1 + (0.607 + 0.794i)T \)
47 \( 1 + (-0.354 - 0.935i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.485 + 0.873i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.607 + 0.794i)T \)
71 \( 1 + (0.885 + 0.464i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.568 + 0.822i)T \)
83 \( 1 + (-0.399 + 0.916i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.0724 + 0.997i)T \)
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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

−20.10137613315713528350880608852, −19.4051761473141908766763406601, −18.8017581711189760338574265144, −17.77383854673442456383495333029, −17.49315712617622784337375186893, −16.89625075065011814871573264307, −16.143032140179072211634378175586, −14.44569380447957270154728415250, −14.08318410284689447663908978839, −13.23959824415283345914947890279, −12.32820328337106285972656600446, −12.043237648070796221532231515834, −10.84759523224200167396578049345, −10.16252484588371153066372285821, −9.52575362238374057824490933377, −8.28644723658541849986212889814, −8.03491753385664151712976810103, −6.9853647371074397520376410749, −6.31416602553859577717903412357, −5.17393638970602602535464308666, −4.04532410648909100711160084950, −2.90541652970905242084778170024, −1.88533227300218546947700888263, −1.465259083967919021552735946160, −0.48658573407621297257148550099, 0.84511399789866658253353334512, 2.3913771224625922002335003280, 2.64876955517874833189181113147, 4.441478747399961250023724903736, 5.308346085940393530534258838600, 5.66900220121595869066566185827, 6.67224866936076381700323256766, 7.72264022043301581865044909881, 8.60233794517483517365620590461, 9.298696893134415014381008470629, 9.89982578393703402681064086152, 10.49414980735916101374666654170, 11.39252804667375842302834138598, 12.19802120857167631950532776944, 13.69020433409756904120160082383, 14.344607858979836369636395611146, 14.88147949813510856251583108402, 15.63066913477807222298823935370, 16.27987393843727041631536988076, 17.25875658328845360105730731831, 17.65604567838074371578102578028, 18.31423934466477460036267852927, 19.30297458532359956683807791958, 20.03212156176436365524003753220, 20.98995809290335420143347088318

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