↑

Properties

Label 2-465-15.2-c1-0-36
Degree $2$
Conductor $465$
Sign $0.0166 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.523 + 0.523i)2-s + (1.58 + 0.695i)3-s + 1.45i·4-s + (2.20 + 0.349i)5-s + (−1.19 + 0.466i)6-s + (2.00 + 2.00i)7-s + (−1.80 − 1.80i)8-s + (2.03 + 2.20i)9-s + (−1.34 + 0.973i)10-s − 2.51i·11-s + (−1.00 + 2.30i)12-s + (−1.36 + 1.36i)13-s − 2.09·14-s + (3.26 + 2.09i)15-s − 1.00·16-s + (2.74 − 2.74i)17-s + ⋯
L(s)  = 1  + (−0.370 + 0.370i)2-s + (0.915 + 0.401i)3-s + 0.725i·4-s + (0.987 + 0.156i)5-s + (−0.488 + 0.190i)6-s + (0.757 + 0.757i)7-s + (−0.639 − 0.639i)8-s + (0.677 + 0.735i)9-s + (−0.423 + 0.307i)10-s − 0.756i·11-s + (−0.291 + 0.664i)12-s + (−0.378 + 0.378i)13-s − 0.560·14-s + (0.841 + 0.539i)15-s − 0.251·16-s + (0.666 − 0.666i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0166 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0166 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.0166 - 0.999i$
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: \((2,\ 465,\ (\ :1/2),\ 0.0166 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34791 + 1.32562i\)
\(L(\frac12)\) \(\approx\) \(1.34791 + 1.32562i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 - 0.695i)T \)
5 \( 1 + (-2.20 - 0.349i)T \)
31 \( 1 + T \)
good2 \( 1 + (0.523 - 0.523i)T - 2iT^{2} \)
7 \( 1 + (-2.00 - 2.00i)T + 7iT^{2} \)
11 \( 1 + 2.51iT - 11T^{2} \)
13 \( 1 + (1.36 - 1.36i)T - 13iT^{2} \)
17 \( 1 + (-2.74 + 2.74i)T - 17iT^{2} \)
19 \( 1 + 4.33iT - 19T^{2} \)
23 \( 1 + (6.46 + 6.46i)T + 23iT^{2} \)
29 \( 1 + 5.35T + 29T^{2} \)
37 \( 1 + (1.27 + 1.27i)T + 37iT^{2} \)
41 \( 1 - 0.543iT - 41T^{2} \)
43 \( 1 + (4.32 - 4.32i)T - 43iT^{2} \)
47 \( 1 + (1.52 - 1.52i)T - 47iT^{2} \)
53 \( 1 + (-6.26 - 6.26i)T + 53iT^{2} \)
59 \( 1 - 3.16T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + (-7.77 - 7.77i)T + 67iT^{2} \)
71 \( 1 + 7.13iT - 71T^{2} \)
73 \( 1 + (6.43 - 6.43i)T - 73iT^{2} \)
79 \( 1 + 7.24iT - 79T^{2} \)
83 \( 1 + (-5.53 - 5.53i)T + 83iT^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-10.2 - 10.2i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.18407838997814622025099078613, −10.03374214265475701707963264061, −9.205850167902340530489714147831, −8.660303832105238973322559177580, −7.85956726158535039224767265774, −6.85670941288060902498453699521, −5.63822188483632317895757807925, −4.47609392544767366804502649097, −3.04844440154799839373938873290, −2.18344686729432628491950594491, 1.48166639968611639519251352562, 2.02284018357410137265055902805, 3.73687344481090990288782244901, 5.16522874021691140623249810221, 6.11168428383724760791560817277, 7.38937575837064434859818186698, 8.147497558909810062199731797632, 9.227994092069756342139452667082, 10.08849629723712812348460496298, 10.26631948774760639259393887936

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