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Properties

Label 1-1472-1472.229-r1-0-0
Degree $1$
Conductor $1472$
Sign $-0.995 - 0.0980i$
Analytic cond. $158.188$
Root an. cond. $158.188$
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  + (−0.382 − 0.923i)3-s + (0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.923 + 0.382i)13-s − i·15-s − i·17-s + (−0.923 + 0.382i)19-s + (0.382 − 0.923i)21-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (0.382 + 0.923i)29-s − 31-s + 33-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)3-s + (0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.923 + 0.382i)13-s − i·15-s − i·17-s + (−0.923 + 0.382i)19-s + (0.382 − 0.923i)21-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (0.382 + 0.923i)29-s − 31-s + 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $-0.995 - 0.0980i$
Analytic conductor: \(158.188\)
Root analytic conductor: \(158.188\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1472,\ (1:\ ),\ -0.995 - 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02283845520 + 0.4648874355i\)
\(L(\frac12)\) \(\approx\) \(0.02283845520 + 0.4648874355i\)
\(L(1)\) \(\approx\) \(0.9194964356 + 0.08537392023i\)
\(L(1)\) \(\approx\) \(0.9194964356 + 0.08537392023i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.382 - 0.923i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.923 + 0.382i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.382 - 0.923i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (-0.923 - 0.382i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 + (0.382 + 0.923i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.923 + 0.382i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 + T \)
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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

−20.3201523039212997431240237134, −19.5647348391014363297504263589, −18.3664218339535677283841926941, −17.53459335928161277855074223173, −17.17030283636118995993513611244, −16.42739501756628818319695833284, −15.73551413219801511229927950182, −14.66481423304659249253942315679, −14.17270331634143838897444459165, −13.35272413567800470362813735527, −12.483130227920050147979774338782, −11.38692747070939436110663038525, −10.85879928904245190197591585601, −10.074676462186098595010554311086, −9.43902384674799899108390924702, −8.58711273901169158201325119931, −7.7146961154773845000707939726, −6.54759477407648719342732666362, −5.7060806652244473982193724285, −4.91832387629215200752166148766, −4.47097944561654185463754673706, −3.19709155986031735653916875475, −2.31538305486104618542891074694, −0.9541794929671753883152083415, −0.09374587100710976254812818618, 1.71126699312696083010647550173, 1.90601046817269251763498320119, 2.78531637258987443364618276139, 4.37850797204201810182828207878, 5.374103052848471940058171641413, 5.81890104059041386351189454089, 6.889493214488791761179396031109, 7.388474254706020984039169119692, 8.466094286965919946890629799691, 9.158308814101096881849975760009, 10.38640779012982185868217348869, 10.761770566035169946322696165843, 11.94754311390091813389129962420, 12.51665622014219500369315768297, 13.056867736131312013779733293971, 14.27521378814795409803796129442, 14.52925382731349795279808584012, 15.41945548962643786317873397056, 16.722543268886140338427905559010, 17.40119001336108954092383341884, 17.76088208209420214346772721235, 18.58617644801517866601627595942, 19.102842067496640487183455190231, 20.057539268323852514243568616866, 20.997213658824409776773188846231

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