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Properties

Label 1-455-455.237-r0-0-0
Degree $1$
Conductor $455$
Sign $-0.974 + 0.223i$
Analytic cond. $2.11301$
Root an. cond. $2.11301$
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  + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s − i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s − i·12-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s − i·18-s + (−0.5 − 0.866i)19-s + (0.866 − 0.5i)22-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s − i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s − i·12-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s − i·18-s + (−0.5 − 0.866i)19-s + (0.866 − 0.5i)22-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(455\)    =    \(5 \cdot 7 \cdot 13\)
Sign: $-0.974 + 0.223i$
Analytic conductor: \(2.11301\)
Root analytic conductor: \(2.11301\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{455} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 455,\ (0:\ ),\ -0.974 + 0.223i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01291424269 - 0.1140978882i\)
\(L(\frac12)\) \(\approx\) \(0.01291424269 - 0.1140978882i\)
\(L(1)\) \(\approx\) \(0.4072960408 - 0.1335041898i\)
\(L(1)\) \(\approx\) \(0.4072960408 - 0.1335041898i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−24.38718070984107792652794730701, −23.64454144429573306604187270400, −22.9123301966603891676544000416, −21.877294767651385820215123790616, −20.956503533643887960120709260814, −20.13872253696386103064902860203, −18.90080134349442228181144547645, −18.33076652435238431986824001710, −17.4401826348131168808723344244, −16.59408786749421111462542433061, −16.06305239366772520345364895640, −15.20461515020980877008404729140, −14.27975110724281266216230786774, −12.97445542056540038767864378768, −11.77150530464772411775511379423, −10.8612745322483268047637410835, −10.40450670411846618092271613490, −9.22814734787404807861071899510, −8.49231326924408326266377625784, −7.22217458616934288737144267637, −6.34462341910297701969986902927, −5.500947242469187215144384693727, −4.56669003474627046025410551378, −2.9913550592757968789052623723, −1.35329261192508956265384533808, 0.09955720614714503013722161627, 1.63751173096813664976762374386, 2.495040021592334035041487237522, 4.10365909964949535729408595857, 5.235835064827768521377287890727, 6.63753839282094906356956420341, 7.204057123525701663210876921509, 8.254036667357257503489007458795, 9.3243057573375668095675397457, 10.407476653878317859018808322608, 11.02618895368857055492511506959, 11.905412234772858647864215781072, 12.83077741033661877344019672473, 13.35018294668188253517052372181, 15.16405410727237765387789789507, 15.90618869916954622889960733967, 16.99897056312205291775315814838, 17.597121308109660534841558588239, 18.16880485835887276723570336608, 19.21293975128420145912745938235, 19.78344327709684310973326661538, 20.942789371032187901047676702102, 21.70964843567493366102154394231, 22.58901358175818727597041393875, 23.49781325964721480966767959365

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