Properties

Label 2-465-465.254-c1-0-14
Degree $2$
Conductor $465$
Sign $-0.147 - 0.989i$
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  + 1.36·2-s + (−1.53 − 0.805i)3-s − 0.137·4-s + (−0.288 + 2.21i)5-s + (−2.09 − 1.09i)6-s + (1.46 − 0.845i)7-s − 2.91·8-s + (1.70 + 2.47i)9-s + (−0.394 + 3.02i)10-s + (−2.40 + 4.16i)11-s + (0.210 + 0.110i)12-s + (0.307 − 0.532i)13-s + (1.99 − 1.15i)14-s + (2.22 − 3.16i)15-s − 3.70·16-s + (−3.77 + 2.17i)17-s + ⋯
L(s)  = 1  + 0.964·2-s + (−0.885 − 0.465i)3-s − 0.0688·4-s + (−0.129 + 0.991i)5-s + (−0.854 − 0.448i)6-s + (0.553 − 0.319i)7-s − 1.03·8-s + (0.567 + 0.823i)9-s + (−0.124 + 0.956i)10-s + (−0.725 + 1.25i)11-s + (0.0609 + 0.0320i)12-s + (0.0852 − 0.147i)13-s + (0.534 − 0.308i)14-s + (0.575 − 0.817i)15-s − 0.926·16-s + (−0.914 + 0.528i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.147 - 0.989i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.147 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.699170 + 0.811321i\)
\(L(\frac12)\) \(\approx\) \(0.699170 + 0.811321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.53 + 0.805i)T \)
5 \( 1 + (0.288 - 2.21i)T \)
31 \( 1 + (0.0385 - 5.56i)T \)
good2 \( 1 - 1.36T + 2T^{2} \)
7 \( 1 + (-1.46 + 0.845i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.40 - 4.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.307 + 0.532i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.77 - 2.17i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.90 - 5.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.43iT - 23T^{2} \)
29 \( 1 - 8.46T + 29T^{2} \)
37 \( 1 + (2.81 + 4.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.07 + 4.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.53 + 4.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.15T + 47T^{2} \)
53 \( 1 + (5.16 + 2.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.93 + 2.84i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 9.69iT - 61T^{2} \)
67 \( 1 + (-4.16 - 2.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.23 - 3.59i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.805 - 1.39i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.206 + 0.119i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.80 + 1.04i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−11.48802766257596092046496932247, −10.54145772026307670967527614562, −9.939321928936817401383604171914, −8.275674958237203700019234192159, −7.29915113482927060978460310279, −6.55501745832513730663642930623, −5.47131889932637240035254710471, −4.71883873378958621890396603786, −3.59808946396191481836724507317, −2.03533354885596130975782830185, 0.53959059149828436034502664448, 2.99132872709275049920065239072, 4.46451590139728330469905576649, 4.89036047534593491840999946563, 5.67222127945606389752354170427, 6.63629494518667635602569383902, 8.355196052847217036632682769647, 8.900591080881748608873786568614, 9.960767074340895060517976403811, 11.28211701462832218898562140630

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