Properties

Label 1-460-460.239-r1-0-0
Degree $1$
Conductor $460$
Sign $0.0230 + 0.999i$
Analytic cond. $49.4338$
Root an. cond. $49.4338$
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  + (−0.142 + 0.989i)3-s + (−0.654 − 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (0.654 − 0.755i)13-s + (−0.415 − 0.909i)17-s + (−0.415 + 0.909i)19-s + (0.841 − 0.540i)21-s + (0.415 − 0.909i)27-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (0.654 − 0.755i)33-s + (0.959 + 0.281i)37-s + (0.654 + 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)3-s + (−0.654 − 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (0.654 − 0.755i)13-s + (−0.415 − 0.909i)17-s + (−0.415 + 0.909i)19-s + (0.841 − 0.540i)21-s + (0.415 − 0.909i)27-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (0.654 − 0.755i)33-s + (0.959 + 0.281i)37-s + (0.654 + 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0230 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0230 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.0230 + 0.999i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ 0.0230 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7598297271 + 0.7424990268i\)
\(L(\frac12)\) \(\approx\) \(0.7598297271 + 0.7424990268i\)
\(L(1)\) \(\approx\) \(0.8136269076 + 0.1923441082i\)
\(L(1)\) \(\approx\) \(0.8136269076 + 0.1923441082i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (0.959 + 0.281i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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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

−23.67827490007080859964785401613, −22.76242722637417410361007093125, −21.91037597001249738031896235246, −20.959834208713311570900084056249, −19.852097839133015076439366004757, −19.080895607042604766694228475985, −18.46710302851130431561383111977, −17.62943444225891027627771635985, −16.72721726148868766331240297047, −15.62200392993535462709332557077, −14.92248454244403134194024443550, −13.48252742357282823567398120758, −13.13543322081278502542029253464, −12.15588023171955059219780439813, −11.36902807069403553286528194828, −10.267550283477744075820871328209, −9.016806881335112149941937088360, −8.3058505432422109249086452006, −7.14635952584702855512651063421, −6.32493723556579197235354341250, −5.54230952644771907041441290099, −4.1578652653622625351710656166, −2.64648460761720537266803723355, −1.980305201935795849032532037085, −0.38124769782622701205943841039, 0.807328253470726698493779023230, 2.85483416554726988404109653851, 3.54401471709577000752707598187, 4.671009658616505561102652422581, 5.64330502999655198509654599216, 6.61231163000596507930158949475, 7.93880905575278581950712905089, 8.828804077167266458511763204806, 9.97322556185051704237956995361, 10.53496045006976004110529017874, 11.30497301050804897615798975120, 12.60976001691156453385705460835, 13.53256306326147322424871283859, 14.35974193287108627214472115263, 15.5460693895666426317387258703, 16.10480264282677095974678104822, 16.7716978791989036956849974312, 17.84811194202568326165717030595, 18.73649998497422762994800453868, 20.023373748683815795501392211235, 20.43340373816436003403578813943, 21.38608315820986680173223293254, 22.18739647197376434199576879705, 23.22970419404416917196161410517, 23.40303927356154674103262352900

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