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Properties

Label 1-460-460.87-r0-0-0
Degree $1$
Conductor $460$
Sign $0.838 + 0.545i$
Analytic cond. $2.13623$
Root an. cond. $2.13623$
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  + (0.989 − 0.142i)3-s + (0.755 + 0.654i)7-s + (0.959 − 0.281i)9-s + (−0.841 + 0.540i)11-s + (−0.755 + 0.654i)13-s + (0.909 + 0.415i)17-s + (0.415 + 0.909i)19-s + (0.841 + 0.540i)21-s + (0.909 − 0.415i)27-s + (−0.415 + 0.909i)29-s + (0.142 − 0.989i)31-s + (−0.755 + 0.654i)33-s + (−0.281 − 0.959i)37-s + (−0.654 + 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)3-s + (0.755 + 0.654i)7-s + (0.959 − 0.281i)9-s + (−0.841 + 0.540i)11-s + (−0.755 + 0.654i)13-s + (0.909 + 0.415i)17-s + (0.415 + 0.909i)19-s + (0.841 + 0.540i)21-s + (0.909 − 0.415i)27-s + (−0.415 + 0.909i)29-s + (0.142 − 0.989i)31-s + (−0.755 + 0.654i)33-s + (−0.281 − 0.959i)37-s + (−0.654 + 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.838 + 0.545i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ 0.838 + 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.895082731 + 0.5620566761i\)
\(L(\frac12)\) \(\approx\) \(1.895082731 + 0.5620566761i\)
\(L(1)\) \(\approx\) \(1.505087301 + 0.1880821731i\)
\(L(1)\) \(\approx\) \(1.505087301 + 0.1880821731i\)

Euler product

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

−24.09801797682359787367623129178, −23.09109432837427893157963498394, −21.93088094068039462463893649810, −21.08871785602042832024094903659, −20.496761213860406484070953598688, −19.69809217868086841175073133864, −18.80868282436436320847171153367, −17.92101236599475049356925914959, −16.95871060102626544437314048516, −15.8740368578133438922113981329, −15.12861973921411035418854065522, −14.1835873806371859029402871840, −13.600410414223037231531148216819, −12.66695975089523688854334143088, −11.42686001940223003657336532735, −10.39212244556985106993720909428, −9.72527949969075735439034416448, −8.477712622223390577608825568145, −7.78679418677943985647333669299, −7.09121572444962812718990911748, −5.37305509292115554638555752974, −4.59781698981802415092266003773, −3.31532408919937542763790445115, −2.52306104026798590603606342656, −1.08049686629898460660198636063, 1.66198026259436174429242435314, 2.37914360858299841209674304687, 3.594652038982112378698552928669, 4.74750905228348757923552163846, 5.731068044269496181337460188816, 7.28861458857378682489915714681, 7.81718568248863544915016859815, 8.797915776106643377996619046628, 9.68918250925483869469340088658, 10.57405670026356359045923752170, 12.01218748453274001593957651962, 12.526326691396070699481782105811, 13.70138163885998599524099991467, 14.58784480662314883415108285688, 15.041769047422004716973783557165, 16.072026029884218746183977529913, 17.16994763703765259874968995805, 18.38245532792464106036621756870, 18.69596340994033104140592510914, 19.76996396888361938054245139101, 20.73162153306398158321131651082, 21.19732603706509134404339185685, 22.09130646013192460844899789985, 23.36869533228662279585285906433, 24.17304270155816365543697708949

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