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Properties

Label 1-449-449.84-r1-0-0
Degree $1$
Conductor $449$
Sign $0.914 - 0.403i$
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  + (−0.980 − 0.195i)2-s + (0.0980 + 0.995i)3-s + (0.923 + 0.382i)4-s − 5-s + (0.0980 − 0.995i)6-s + (−0.382 + 0.923i)7-s + (−0.831 − 0.555i)8-s + (−0.980 + 0.195i)9-s + (0.980 + 0.195i)10-s + (0.707 − 0.707i)11-s + (−0.290 + 0.956i)12-s + (0.956 − 0.290i)13-s + (0.555 − 0.831i)14-s + (−0.0980 − 0.995i)15-s + (0.707 + 0.707i)16-s + (−0.471 + 0.881i)17-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)2-s + (0.0980 + 0.995i)3-s + (0.923 + 0.382i)4-s − 5-s + (0.0980 − 0.995i)6-s + (−0.382 + 0.923i)7-s + (−0.831 − 0.555i)8-s + (−0.980 + 0.195i)9-s + (0.980 + 0.195i)10-s + (0.707 − 0.707i)11-s + (−0.290 + 0.956i)12-s + (0.956 − 0.290i)13-s + (0.555 − 0.831i)14-s + (−0.0980 − 0.995i)15-s + (0.707 + 0.707i)16-s + (−0.471 + 0.881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $0.914 - 0.403i$
Analytic conductor: \(48.2517\)
Root analytic conductor: \(48.2517\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (1:\ ),\ 0.914 - 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4485871351 - 0.09456747713i\)
\(L(\frac12)\) \(\approx\) \(0.4485871351 - 0.09456747713i\)
\(L(1)\) \(\approx\) \(0.5002226818 + 0.1535841612i\)
\(L(1)\) \(\approx\) \(0.5002226818 + 0.1535841612i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (-0.980 - 0.195i)T \)
3 \( 1 + (0.0980 + 0.995i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.382 + 0.923i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (0.956 - 0.290i)T \)
17 \( 1 + (-0.471 + 0.881i)T \)
19 \( 1 + (-0.995 + 0.0980i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (-0.773 - 0.634i)T \)
31 \( 1 + (-0.773 + 0.634i)T \)
37 \( 1 + (-0.290 - 0.956i)T \)
41 \( 1 + (-0.555 - 0.831i)T \)
43 \( 1 + (0.956 - 0.290i)T \)
47 \( 1 + (-0.995 - 0.0980i)T \)
53 \( 1 + (0.831 - 0.555i)T \)
59 \( 1 + (-0.195 - 0.980i)T \)
61 \( 1 + (-0.831 - 0.555i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.956 + 0.290i)T \)
73 \( 1 + (-0.773 - 0.634i)T \)
79 \( 1 + (0.0980 + 0.995i)T \)
83 \( 1 + (0.634 - 0.773i)T \)
89 \( 1 + (0.923 + 0.382i)T \)
97 \( 1 + (0.980 + 0.195i)T \)
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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

−23.985359820787836879051947132770, −23.22427726185297810752762671511, −22.58413163437190305504937687968, −20.62193493375398181046953494043, −20.12377959398897133521320279516, −19.518921346318889170921122516079, −18.63891161438914154296162223060, −18.030018865511117614901154665466, −16.8533253211026448480396468734, −16.43591395160739853263132679537, −15.20668683889118529373586574529, −14.4069052760068559077912468972, −13.2361005791634985768546736561, −12.23879615996938269192970615287, −11.37774512483564619429585209760, −10.680018893894602826808423232488, −9.29009476197774006454728711293, −8.501188582518239169429791663379, −7.55086783986747287432613306900, −6.890025079766096954368305778340, −6.26265349523678981770797660231, −4.39423331853482850148536155930, −3.17808154475961015585823062886, −1.80196226671940491321069248113, −0.72946091536649757569849308385, 0.24540709808041135469617273466, 2.02295076527313231519868621707, 3.49223355330512530874044349730, 3.80255429802799550099420291727, 5.63783076984380763132349796833, 6.48166082917006704012066812754, 7.9818144237864475984826802594, 8.73334364603057989121524686069, 9.170085862587853719366521924370, 10.511607106313221832683754586668, 11.15280449991043252179559905169, 11.86836400343683458201740318367, 12.93123055688662221238856543534, 14.600991550461762346511035190321, 15.44056963550051781086146205061, 15.89037465990381701418420759781, 16.68719380437955052803646737814, 17.634380054472271155482522498111, 18.859480581694770047480205455902, 19.40659848502749614507752371303, 20.09238598729491699539073756163, 21.163719418188033119078184471825, 21.77021824301895719874749544043, 22.68442240385984742119446722601, 23.81581461081625094766172063856

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