↑

Properties

Label 1-449-449.63-r0-0-0
Degree $1$
Conductor $449$
Sign $0.272 + 0.962i$
Analytic cond. $2.08514$
Root an. cond. $2.08514$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.875 + 0.483i)2-s + (0.799 + 0.601i)3-s + (0.532 + 0.846i)4-s + (0.623 − 0.781i)5-s + (0.408 + 0.912i)6-s + (−0.993 − 0.111i)7-s + (0.0560 + 0.998i)8-s + (0.276 + 0.960i)9-s + (0.923 − 0.382i)10-s + (0.781 + 0.623i)11-s + (−0.0840 + 0.996i)12-s + (0.356 − 0.934i)13-s + (−0.815 − 0.578i)14-s + (0.968 − 0.249i)15-s + (−0.433 + 0.900i)16-s + (−0.686 − 0.726i)17-s + ⋯
L(s)  = 1  + (0.875 + 0.483i)2-s + (0.799 + 0.601i)3-s + (0.532 + 0.846i)4-s + (0.623 − 0.781i)5-s + (0.408 + 0.912i)6-s + (−0.993 − 0.111i)7-s + (0.0560 + 0.998i)8-s + (0.276 + 0.960i)9-s + (0.923 − 0.382i)10-s + (0.781 + 0.623i)11-s + (−0.0840 + 0.996i)12-s + (0.356 − 0.934i)13-s + (−0.815 − 0.578i)14-s + (0.968 − 0.249i)15-s + (−0.433 + 0.900i)16-s + (−0.686 − 0.726i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $0.272 + 0.962i$
Analytic conductor: \(2.08514\)
Root analytic conductor: \(2.08514\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (0:\ ),\ 0.272 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.415603516 + 1.826856251i\)
\(L(\frac12)\) \(\approx\) \(2.415603516 + 1.826856251i\)
\(L(1)\) \(\approx\) \(2.008257986 + 0.9628951580i\)
\(L(1)\) \(\approx\) \(2.008257986 + 0.9628951580i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (0.875 + 0.483i)T \)
3 \( 1 + (0.799 + 0.601i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (-0.993 - 0.111i)T \)
11 \( 1 + (0.781 + 0.623i)T \)
13 \( 1 + (0.356 - 0.934i)T \)
17 \( 1 + (-0.686 - 0.726i)T \)
19 \( 1 + (0.458 + 0.888i)T \)
23 \( 1 + (0.846 + 0.532i)T \)
29 \( 1 + (-0.645 - 0.764i)T \)
31 \( 1 + (-0.645 + 0.764i)T \)
37 \( 1 + (-0.831 - 0.555i)T \)
41 \( 1 + (0.985 - 0.167i)T \)
43 \( 1 + (0.0840 + 0.996i)T \)
47 \( 1 + (-0.408 - 0.912i)T \)
53 \( 1 + (-0.985 + 0.167i)T \)
59 \( 1 + (0.578 - 0.815i)T \)
61 \( 1 + (-0.483 - 0.875i)T \)
67 \( 1 - T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (0.888 - 0.458i)T \)
79 \( 1 + (-0.980 - 0.195i)T \)
83 \( 1 + (0.601 - 0.799i)T \)
89 \( 1 + (-0.330 + 0.943i)T \)
97 \( 1 + (-0.998 - 0.0560i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.99843546796645462379120014038, −22.73023127160861466918283987066, −22.12087229838549221845115672338, −21.393657351983642123735693320, −20.41544732556415991109604595505, −19.38363014915114586220119647608, −19.082594184201639122760300204559, −18.20746406632636241314090846338, −16.84573737162839639615311598241, −15.64338322166421830430241422382, −14.78117916031428790356970205198, −14.04598116878218932098721724452, −13.361723879583650962932674174504, −12.73784878803065527387479372960, −11.54027353955520595329072378362, −10.720908380812786675143352256298, −9.42968855695228134735223858655, −8.993832664680071220292949984159, −7.02400249562719318142535787794, −6.6312221718294680857125564138, −5.802735967741317361695469828962, −4.0590827325365387415281881436, −3.24421316289132360005424359504, −2.43797323855960704915616488196, −1.37366844426912624283071297192, 1.820466230178594744565404804069, 3.065922764579254330385457015959, 3.86533410432655775522442939137, 4.91112178534013498613430487316, 5.77805582510094642535053446359, 6.925700725874472336726021899811, 7.95578960170904088669084311640, 9.10359751629664541331526194408, 9.63962356193348098736103880589, 10.88568046843042446188180559618, 12.310066361248089703229391969027, 13.03577324046362645214656277000, 13.68191704460763398788217220755, 14.54153892735004601640199339737, 15.54180992300062842460689385036, 16.13922791155826098812491730170, 16.91692952623011792533166257029, 17.87197566646716066683756825595, 19.470155730126072853696602691402, 20.25190047470620847255040127783, 20.72829738815926088273298754547, 21.66472207956212935121451496446, 22.558512127728914620354901370007, 23.0061784526347112094835200644, 24.58562575316993848658178735340

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