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Properties

Label 1-449-449.105-r1-0-0
Degree $1$
Conductor $449$
Sign $0.764 + 0.645i$
Analytic cond. $48.2517$
Root an. cond. $48.2517$
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  + (0.934 − 0.356i)2-s + (−0.395 + 0.918i)3-s + (0.745 − 0.666i)4-s + (0.222 + 0.974i)5-s + (−0.0420 + 0.999i)6-s + (0.578 − 0.815i)7-s + (0.458 − 0.888i)8-s + (−0.686 − 0.726i)9-s + (0.555 + 0.831i)10-s + (0.846 + 0.532i)11-s + (0.317 + 0.948i)12-s + (0.343 + 0.939i)13-s + (0.249 − 0.968i)14-s + (−0.983 − 0.181i)15-s + (0.111 − 0.993i)16-s + (0.236 + 0.971i)17-s + ⋯
L(s)  = 1  + (0.934 − 0.356i)2-s + (−0.395 + 0.918i)3-s + (0.745 − 0.666i)4-s + (0.222 + 0.974i)5-s + (−0.0420 + 0.999i)6-s + (0.578 − 0.815i)7-s + (0.458 − 0.888i)8-s + (−0.686 − 0.726i)9-s + (0.555 + 0.831i)10-s + (0.846 + 0.532i)11-s + (0.317 + 0.948i)12-s + (0.343 + 0.939i)13-s + (0.249 − 0.968i)14-s + (−0.983 − 0.181i)15-s + (0.111 − 0.993i)16-s + (0.236 + 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $0.764 + 0.645i$
Analytic conductor: \(48.2517\)
Root analytic conductor: \(48.2517\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (1:\ ),\ 0.764 + 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.780028787 + 1.382354818i\)
\(L(\frac12)\) \(\approx\) \(3.780028787 + 1.382354818i\)
\(L(1)\) \(\approx\) \(1.983698214 + 0.3247182097i\)
\(L(1)\) \(\approx\) \(1.983698214 + 0.3247182097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (0.934 - 0.356i)T \)
3 \( 1 + (-0.395 + 0.918i)T \)
5 \( 1 + (0.222 + 0.974i)T \)
7 \( 1 + (0.578 - 0.815i)T \)
11 \( 1 + (0.846 + 0.532i)T \)
13 \( 1 + (0.343 + 0.939i)T \)
17 \( 1 + (0.236 + 0.971i)T \)
19 \( 1 + (-0.263 - 0.964i)T \)
23 \( 1 + (0.998 - 0.0560i)T \)
29 \( 1 + (0.0700 - 0.997i)T \)
31 \( 1 + (0.0700 + 0.997i)T \)
37 \( 1 + (-0.995 - 0.0980i)T \)
41 \( 1 + (0.601 - 0.799i)T \)
43 \( 1 + (0.948 + 0.317i)T \)
47 \( 1 + (0.999 + 0.0420i)T \)
53 \( 1 + (0.799 + 0.601i)T \)
59 \( 1 + (0.508 + 0.861i)T \)
61 \( 1 + (-0.408 - 0.912i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.0980 + 0.995i)T \)
73 \( 1 + (-0.495 + 0.868i)T \)
79 \( 1 + (0.471 + 0.881i)T \)
83 \( 1 + (-0.929 + 0.369i)T \)
89 \( 1 + (0.483 + 0.875i)T \)
97 \( 1 + (-0.303 + 0.952i)T \)
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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

−23.87280890566994868411631046519, −22.85191319157377621789438800816, −22.25812319730967792665644676991, −21.13689421924141807342698705759, −20.53705820487376355942290771847, −19.49665928278470965574989201462, −18.39843517827105408544704730179, −17.45457345762096247564953923544, −16.752649519375590483336790072804, −15.96150831453607142106713104436, −14.79043817036654831996127474818, −13.94325807538083514430481162200, −13.11423871937817492785506029392, −12.32047670069256606055151481823, −11.79240577732389855890757295621, −10.8505143165671526547296032608, −8.98351593802478697418174017191, −8.25169190199324454462906827422, −7.34630059008757570394270758560, −6.0425722233784474939980664879, −5.56999682311940280249070431191, −4.71006480477844345040926777821, −3.21310208839596084645615104934, −1.97337589962010895900852723227, −0.934785886660113757405869919851, 1.20861342873260529820827075198, 2.54112387438785296112068971383, 3.89390010658101258386823413215, 4.22336794147792625157536996903, 5.45243383110372151604911993759, 6.55350716353991552743142664126, 7.124668861599936569333689477448, 8.98468772999199646743731353050, 10.06401111872049273372848957049, 10.82239712742842448398640927420, 11.30001226307403296962718848045, 12.2683688995310517399925943052, 13.67472105772245182689020352403, 14.34811620073808868971287061285, 14.96546420168416565663247512846, 15.78377295246761013257868774289, 17.04351969997024005173353329851, 17.52943682940300708189099771697, 19.042839763991549455636301516511, 19.76080291439682664133754810175, 20.87767486137928278950881220813, 21.38725763191526883692502294895, 22.11883808930721676056014877868, 23.01518789850706921243894143802, 23.40965084831203620477462142138

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