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Properties

Label 1-1441-1441.1043-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.215 + 0.976i$
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  + (−0.958 + 0.285i)2-s + (−0.168 + 0.985i)3-s + (0.836 − 0.548i)4-s + (−0.443 − 0.896i)5-s + (−0.120 − 0.992i)6-s + (0.485 + 0.873i)7-s + (−0.644 + 0.764i)8-s + (−0.943 − 0.331i)9-s + (0.681 + 0.732i)10-s + (0.399 + 0.916i)12-s + (0.981 − 0.192i)13-s + (−0.715 − 0.698i)14-s + (0.958 − 0.285i)15-s + (0.399 − 0.916i)16-s + (0.748 + 0.663i)17-s + (0.998 + 0.0483i)18-s + ⋯
L(s)  = 1  + (−0.958 + 0.285i)2-s + (−0.168 + 0.985i)3-s + (0.836 − 0.548i)4-s + (−0.443 − 0.896i)5-s + (−0.120 − 0.992i)6-s + (0.485 + 0.873i)7-s + (−0.644 + 0.764i)8-s + (−0.943 − 0.331i)9-s + (0.681 + 0.732i)10-s + (0.399 + 0.916i)12-s + (0.981 − 0.192i)13-s + (−0.715 − 0.698i)14-s + (0.958 − 0.285i)15-s + (0.399 − 0.916i)16-s + (0.748 + 0.663i)17-s + (0.998 + 0.0483i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.215 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5903240968 + 0.7344691750i\)
\(L(\frac12)\) \(\approx\) \(0.5903240968 + 0.7344691750i\)
\(L(1)\) \(\approx\) \(0.6029431878 + 0.2273300131i\)
\(L(1)\) \(\approx\) \(0.6029431878 + 0.2273300131i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.958 + 0.285i)T \)
3 \( 1 + (-0.168 + 0.985i)T \)
5 \( 1 + (-0.443 - 0.896i)T \)
7 \( 1 + (0.485 + 0.873i)T \)
13 \( 1 + (0.981 - 0.192i)T \)
17 \( 1 + (0.748 + 0.663i)T \)
19 \( 1 + (-0.399 - 0.916i)T \)
23 \( 1 + (-0.644 - 0.764i)T \)
29 \( 1 + (-0.0241 - 0.999i)T \)
31 \( 1 + (-0.644 + 0.764i)T \)
37 \( 1 + (0.262 + 0.964i)T \)
41 \( 1 + (-0.748 + 0.663i)T \)
43 \( 1 + (0.715 - 0.698i)T \)
47 \( 1 + (-0.958 + 0.285i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.215 + 0.976i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.715 - 0.698i)T \)
71 \( 1 + (-0.644 + 0.764i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.443 + 0.896i)T \)
83 \( 1 + (0.262 - 0.964i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.485 - 0.873i)T \)
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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

−20.21071154047500768866319490717, −19.268344919504703158855540069346, −18.85680420423311180480962071990, −17.96570192203936002517991076814, −17.81668742835758343998340980232, −16.5577788627786094080848291724, −16.253579953992994877293119367527, −14.94218263434884769570006507930, −14.21451949864635172878705440805, −13.461237219350587962628839548087, −12.413485131622966912219883562374, −11.691198174015328524897704739023, −11.02981878526277665028449049248, −10.559146468548544016636578694614, −9.51823775605328891188345182704, −8.35582422055496036776893099017, −7.759165385643385950193032282038, −7.251013670947987167999344955706, −6.48103333043011284192781808719, −5.63044130759658401348247798553, −3.89281330144227564929702241118, −3.297543408019387521791008337026, −2.102111732697491940559719509774, −1.35794102319239459783495064494, −0.38298612731927849947875386345, 0.68456487607794465823967983410, 1.775629226402319702779647357059, 2.956266775409348855462285446, 4.07002338722703498767774864417, 5.04420244506193887192182595996, 5.704671932931779349561061131627, 6.48343068420715168045804698333, 7.97726821875605051229598826523, 8.46081986579098994997825532240, 8.94021818488731523961832422006, 9.83092457488395770562802389166, 10.62654315594467270149383670460, 11.48091305366841628600506512910, 11.9210689031930700301568376179, 12.98189997101686769392368096422, 14.323084243177392047744367445279, 15.10700879132628499507658814739, 15.59904913348277023658752248434, 16.25139589538907494781756912082, 16.8859640407016952188753155343, 17.63222847935716541444203315355, 18.37200902492483584532939341664, 19.29819338263308165886469694556, 19.989377617579674214828870728908, 20.84392750996925640532169665447

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