Properties

Label 2-465-465.119-c1-0-25
Degree $2$
Conductor $465$
Sign $0.726 + 0.687i$
Analytic cond. $3.71304$
Root an. cond. $1.92692$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.15·2-s + (0.150 + 1.72i)3-s − 0.658·4-s + (−2.23 + 0.0890i)5-s + (−0.174 − 1.99i)6-s + (−0.189 − 0.109i)7-s + 3.07·8-s + (−2.95 + 0.519i)9-s + (2.58 − 0.103i)10-s + (−3.23 − 5.60i)11-s + (−0.0990 − 1.13i)12-s + (1.94 + 3.37i)13-s + (0.220 + 0.127i)14-s + (−0.489 − 3.84i)15-s − 2.24·16-s + (1.53 + 0.889i)17-s + ⋯
L(s)  = 1  − 0.819·2-s + (0.0868 + 0.996i)3-s − 0.329·4-s + (−0.999 + 0.0398i)5-s + (−0.0711 − 0.815i)6-s + (−0.0718 − 0.0414i)7-s + 1.08·8-s + (−0.984 + 0.173i)9-s + (0.818 − 0.0326i)10-s + (−0.975 − 1.68i)11-s + (−0.0285 − 0.327i)12-s + (0.540 + 0.935i)13-s + (0.0588 + 0.0339i)14-s + (−0.126 − 0.991i)15-s − 0.562·16-s + (0.373 + 0.215i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403854 - 0.160699i\)
\(L(\frac12)\) \(\approx\) \(0.403854 - 0.160699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.150 - 1.72i)T \)
5 \( 1 + (2.23 - 0.0890i)T \)
31 \( 1 + (-5.53 - 0.642i)T \)
good2 \( 1 + 1.15T + 2T^{2} \)
7 \( 1 + (0.189 + 0.109i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.23 + 5.60i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.94 - 3.37i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.53 - 0.889i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.113 + 0.195i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.66iT - 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
37 \( 1 + (1.21 - 2.10i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.613 + 0.354i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.15 + 7.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.68T + 47T^{2} \)
53 \( 1 + (7.94 - 4.58i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.20 + 4.15i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 + (0.362 - 0.209i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.13 + 2.38i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.75 + 8.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.00 + 4.04i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.68 - 1.55i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.11T + 89T^{2} \)
97 \( 1 + 3.68iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.76886636019399868253565589961, −10.13595542410148085292476111782, −8.826609787967564969142197120228, −8.555876907525432019162258790125, −7.77341054379247071758596870978, −6.27293966899897790332749698932, −4.95398959534743597554742308464, −4.08417529943278967089186448076, −3.04157733743927499645393356822, −0.42314019486813436749465693339, 1.17740117538407906078380871358, 2.86294985617475270369117813819, 4.38754059869340980357745106222, 5.50613081723223781726339102572, 7.06984816916554035677340878723, 7.75952455337022767012450686382, 8.103974789234819993544184762513, 9.241046041416487452332046070658, 10.17265688850855217824902041139, 11.06229248313102966349503322788

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