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Properties

Label 2-465-15.2-c1-0-0
Degree $2$
Conductor $465$
Sign $-0.988 + 0.150i$
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  + (−1.67 + 1.67i)2-s + (−0.586 − 1.62i)3-s − 3.62i·4-s + (1.63 + 1.52i)5-s + (3.71 + 1.75i)6-s + (−2.99 − 2.99i)7-s + (2.73 + 2.73i)8-s + (−2.31 + 1.91i)9-s + (−5.30 + 0.196i)10-s − 2.13i·11-s + (−5.91 + 2.12i)12-s + (1.06 − 1.06i)13-s + 10.0·14-s + (1.51 − 3.56i)15-s − 1.91·16-s + (−4.06 + 4.06i)17-s + ⋯
L(s)  = 1  + (−1.18 + 1.18i)2-s + (−0.338 − 0.940i)3-s − 1.81i·4-s + (0.732 + 0.680i)5-s + (1.51 + 0.714i)6-s + (−1.13 − 1.13i)7-s + (0.966 + 0.966i)8-s + (−0.770 + 0.637i)9-s + (−1.67 + 0.0620i)10-s − 0.642i·11-s + (−1.70 + 0.614i)12-s + (0.295 − 0.295i)13-s + 2.68·14-s + (0.392 − 0.919i)15-s − 0.478·16-s + (−0.986 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.988 + 0.150i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.988 + 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00677833 - 0.0898428i\)
\(L(\frac12)\) \(\approx\) \(0.00677833 - 0.0898428i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.586 + 1.62i)T \)
5 \( 1 + (-1.63 - 1.52i)T \)
31 \( 1 + T \)
good2 \( 1 + (1.67 - 1.67i)T - 2iT^{2} \)
7 \( 1 + (2.99 + 2.99i)T + 7iT^{2} \)
11 \( 1 + 2.13iT - 11T^{2} \)
13 \( 1 + (-1.06 + 1.06i)T - 13iT^{2} \)
17 \( 1 + (4.06 - 4.06i)T - 17iT^{2} \)
19 \( 1 - 7.95iT - 19T^{2} \)
23 \( 1 + (1.54 + 1.54i)T + 23iT^{2} \)
29 \( 1 + 5.69T + 29T^{2} \)
37 \( 1 + (4.40 + 4.40i)T + 37iT^{2} \)
41 \( 1 - 2.72iT - 41T^{2} \)
43 \( 1 + (1.63 - 1.63i)T - 43iT^{2} \)
47 \( 1 + (2.99 - 2.99i)T - 47iT^{2} \)
53 \( 1 + (0.906 + 0.906i)T + 53iT^{2} \)
59 \( 1 - 5.63T + 59T^{2} \)
61 \( 1 + 9.56T + 61T^{2} \)
67 \( 1 + (6.08 + 6.08i)T + 67iT^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + (10.6 - 10.6i)T - 73iT^{2} \)
79 \( 1 - 9.68iT - 79T^{2} \)
83 \( 1 + (-1.44 - 1.44i)T + 83iT^{2} \)
89 \( 1 + 6.55T + 89T^{2} \)
97 \( 1 + (-3.25 - 3.25i)T + 97iT^{2} \)
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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

−10.93990106518344067529807742153, −10.47098027063369862903683177979, −9.669790073614339952526991399895, −8.566828298847217529896343585597, −7.69365052216383126488731098873, −6.92908533621068927382296061090, −6.15394961702038556899868834415, −5.86113061771233700342902853468, −3.51036326854384556386259067299, −1.58700561219801813717178461420, 0.082255145644885012035131335441, 2.17407155240356545639596515923, 3.11362643098971655078309078481, 4.62700967120880080431210848566, 5.71374004193262216933952682315, 6.90806475536177061133011647320, 8.830666127964224531784121099093, 9.013383509119655361948251922177, 9.603268766813538414259054460771, 10.30708098069746797297734567422

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