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Properties

Label 1-449-449.139-r1-0-0
Degree $1$
Conductor $449$
Sign $-0.546 - 0.837i$
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  + (−0.990 + 0.139i)2-s + (−0.495 + 0.868i)3-s + (0.960 − 0.276i)4-s + (0.900 − 0.433i)5-s + (0.369 − 0.929i)6-s + (−0.666 − 0.745i)7-s + (−0.912 + 0.408i)8-s + (−0.508 − 0.861i)9-s + (−0.831 + 0.555i)10-s + (0.330 − 0.943i)11-s + (−0.236 + 0.971i)12-s + (0.0140 + 0.999i)13-s + (0.764 + 0.645i)14-s + (−0.0700 + 0.997i)15-s + (0.846 − 0.532i)16-s + (−0.839 + 0.543i)17-s + ⋯
L(s)  = 1  + (−0.990 + 0.139i)2-s + (−0.495 + 0.868i)3-s + (0.960 − 0.276i)4-s + (0.900 − 0.433i)5-s + (0.369 − 0.929i)6-s + (−0.666 − 0.745i)7-s + (−0.912 + 0.408i)8-s + (−0.508 − 0.861i)9-s + (−0.831 + 0.555i)10-s + (0.330 − 0.943i)11-s + (−0.236 + 0.971i)12-s + (0.0140 + 0.999i)13-s + (0.764 + 0.645i)14-s + (−0.0700 + 0.997i)15-s + (0.846 − 0.532i)16-s + (−0.839 + 0.543i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $-0.546 - 0.837i$
Analytic conductor: \(48.2517\)
Root analytic conductor: \(48.2517\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (1:\ ),\ -0.546 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2265968750 - 0.4183425741i\)
\(L(\frac12)\) \(\approx\) \(0.2265968750 - 0.4183425741i\)
\(L(1)\) \(\approx\) \(0.5896088869 + 0.004031847288i\)
\(L(1)\) \(\approx\) \(0.5896088869 + 0.004031847288i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (-0.990 + 0.139i)T \)
3 \( 1 + (-0.495 + 0.868i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (-0.666 - 0.745i)T \)
11 \( 1 + (0.330 - 0.943i)T \)
13 \( 1 + (0.0140 + 0.999i)T \)
17 \( 1 + (-0.839 + 0.543i)T \)
19 \( 1 + (0.676 - 0.736i)T \)
23 \( 1 + (0.875 - 0.483i)T \)
29 \( 1 + (-0.590 + 0.807i)T \)
31 \( 1 + (-0.590 - 0.807i)T \)
37 \( 1 + (0.634 + 0.773i)T \)
41 \( 1 + (-0.458 - 0.888i)T \)
43 \( 1 + (0.971 - 0.236i)T \)
47 \( 1 + (-0.929 - 0.369i)T \)
53 \( 1 + (0.888 - 0.458i)T \)
59 \( 1 + (-0.996 + 0.0840i)T \)
61 \( 1 + (0.601 + 0.799i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.773 - 0.634i)T \)
73 \( 1 + (-0.999 + 0.0420i)T \)
79 \( 1 + (0.956 + 0.290i)T \)
83 \( 1 + (-0.964 + 0.263i)T \)
89 \( 1 + (-0.985 - 0.167i)T \)
97 \( 1 + (-0.356 - 0.934i)T \)
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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

−24.69518049543686785843184581879, −23.01661358172454785119983522276, −22.46873748006116545131171161547, −21.54699544735352902641098363810, −20.36986532125466224676355997788, −19.605265118673564623363833637, −18.64736986165203350718770020037, −18.0250637312612738051702728041, −17.55979034696693171734106183872, −16.61510922805193586042515746791, −15.58288162771680694294827356536, −14.6171199745416825229679417465, −13.20216380706846848306286976668, −12.61950207524871894911514224022, −11.6532494937673644797574713698, −10.74418106076783704612306142861, −9.741331227449259912641694711897, −9.09547644897819081097269946212, −7.75239169155972119031497045831, −6.9601358300637375344907362683, −6.14523186128189908465647655557, −5.35538815804236011641367898658, −3.0647851345532713921391370706, −2.26377174000144803467732471935, −1.27881757084901951379217334776, 0.19981432551292730502224915024, 1.27757083800318532622505784322, 2.82578312943700319059851236234, 4.093843536083896785734027314967, 5.40214519974576819067923358136, 6.32985895826879180443626209074, 6.995087340567365787067576077159, 8.81026112099852332575647743066, 9.13533931006369626007511707558, 10.042486761462725345416251486951, 10.88020582522192963654803542571, 11.55605176673884956599087499738, 12.9491813983203043280906077437, 13.98468875499234353723807076646, 15.04114928097772928055713785710, 16.20863384562758033684271834146, 16.63695126831905416706347731183, 17.17875261566296312247048360916, 18.11490120825507751845077233758, 19.19684434909109473657301387035, 20.12988546367039414680740636840, 20.81454175552921346327907764404, 21.73479458563681835980330274136, 22.369330670309225811933495351915, 23.875033215312981493083151359485

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