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Properties

Label 1-449-449.123-r1-0-0
Degree $1$
Conductor $449$
Sign $0.189 - 0.981i$
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  + (0.799 − 0.601i)2-s + (0.992 − 0.125i)3-s + (0.276 − 0.960i)4-s + (0.900 + 0.433i)5-s + (0.716 − 0.697i)6-s + (0.745 + 0.666i)7-s + (−0.356 − 0.934i)8-s + (0.968 − 0.249i)9-s + (0.980 − 0.195i)10-s + (−0.330 − 0.943i)11-s + (0.153 − 0.988i)12-s + (−0.369 − 0.929i)13-s + (0.996 + 0.0840i)14-s + (0.948 + 0.317i)15-s + (−0.846 − 0.532i)16-s + (0.983 + 0.181i)17-s + ⋯
L(s)  = 1  + (0.799 − 0.601i)2-s + (0.992 − 0.125i)3-s + (0.276 − 0.960i)4-s + (0.900 + 0.433i)5-s + (0.716 − 0.697i)6-s + (0.745 + 0.666i)7-s + (−0.356 − 0.934i)8-s + (0.968 − 0.249i)9-s + (0.980 − 0.195i)10-s + (−0.330 − 0.943i)11-s + (0.153 − 0.988i)12-s + (−0.369 − 0.929i)13-s + (0.996 + 0.0840i)14-s + (0.948 + 0.317i)15-s + (−0.846 − 0.532i)16-s + (0.983 + 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(48.2517\)
Root analytic conductor: \(48.2517\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (1:\ ),\ 0.189 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.722747505 - 3.897024735i\)
\(L(\frac12)\) \(\approx\) \(4.722747505 - 3.897024735i\)
\(L(1)\) \(\approx\) \(2.566512840 - 1.228811409i\)
\(L(1)\) \(\approx\) \(2.566512840 - 1.228811409i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (0.799 - 0.601i)T \)
3 \( 1 + (0.992 - 0.125i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (0.745 + 0.666i)T \)
11 \( 1 + (-0.330 - 0.943i)T \)
13 \( 1 + (-0.369 - 0.929i)T \)
17 \( 1 + (0.983 + 0.181i)T \)
19 \( 1 + (-0.939 + 0.343i)T \)
23 \( 1 + (0.483 - 0.875i)T \)
29 \( 1 + (-0.971 + 0.236i)T \)
31 \( 1 + (-0.971 - 0.236i)T \)
37 \( 1 + (0.290 - 0.956i)T \)
41 \( 1 + (0.303 + 0.952i)T \)
43 \( 1 + (0.988 - 0.153i)T \)
47 \( 1 + (0.697 + 0.716i)T \)
53 \( 1 + (0.952 - 0.303i)T \)
59 \( 1 + (0.645 + 0.764i)T \)
61 \( 1 + (0.990 - 0.139i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.956 + 0.290i)T \)
73 \( 1 + (-0.421 + 0.906i)T \)
79 \( 1 + (-0.0980 + 0.995i)T \)
83 \( 1 + (-0.612 + 0.790i)T \)
89 \( 1 + (0.167 + 0.985i)T \)
97 \( 1 + (-0.408 - 0.912i)T \)
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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

−23.9551742521453286144217570057, −23.47509924254197857405431929313, −22.07585138401167469168574412881, −21.278047128275408599684858270445, −20.79339645485605787224823404582, −20.150823976282883592685396006677, −18.833337748725680024777821501, −17.60959012873100411612164094200, −17.0015952641831990828317134585, −16.0902597257822566103581342932, −14.80938273883655369327071104647, −14.54761304320713332381215884594, −13.53741576473974337704497828905, −13.0279494955922107385632952692, −11.96603404638813909720850425195, −10.58388531125773968182250686347, −9.50760135090814708719409251892, −8.6732677337956179613112898134, −7.518058660463711673250070544612, −7.03238692040907154687955103262, −5.49858821257712645694914580571, −4.64614344169385521368639957034, −3.87502452023072137050054849697, −2.41800606359954444008412997875, −1.652488162610110153344044313266, 1.123678384952100942690301026503, 2.29228693950045225460702650160, 2.816940515129551629957965687473, 3.94934689395118853430020381964, 5.371534182402612078431366796116, 5.93342129489872799562522240971, 7.32689000898990377757851007876, 8.46542232682535692750027173343, 9.424354604524059050100718502778, 10.4255327740975709824525943352, 11.08593283583741088099980697614, 12.599706721577069751223173972183, 12.96960312943325029704205143621, 14.14421395612251278887257985447, 14.6283816081460716072968121921, 15.183023514116719022630870296562, 16.52805319463913352063588956265, 18.00712314802697534114668101834, 18.67100430512618025684784337515, 19.2917741658945654297321803495, 20.53528197752826920400427325355, 21.084757579115025688000780963426, 21.639232803815509230351062812965, 22.46991530236567474526491697817, 23.69158554887710593525222992368

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