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Properties

Label 2-460-460.99-c1-0-66
Degree $2$
Conductor $460$
Sign $-0.433 + 0.901i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (0.926 − 1.06i)2-s + (2.78 − 1.79i)3-s + (−0.284 − 1.97i)4-s + (−2.03 + 0.928i)5-s + (0.667 − 4.64i)6-s + (0.518 + 1.76i)7-s + (−2.37 − 1.52i)8-s + (3.32 − 7.27i)9-s + (−0.890 + 3.03i)10-s + (−4.34 − 5.01i)12-s + (2.36 + 1.08i)14-s + (−4.00 + 6.23i)15-s + (−3.83 + 1.12i)16-s + (−4.69 − 10.2i)18-s + (2.41 + 3.76i)20-s + (4.60 + 3.99i)21-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (1.61 − 1.03i)3-s + (−0.142 − 0.989i)4-s + (−0.909 + 0.415i)5-s + (0.272 − 1.89i)6-s + (0.195 + 0.666i)7-s + (−0.841 − 0.540i)8-s + (1.10 − 2.42i)9-s + (−0.281 + 0.959i)10-s + (−1.25 − 1.44i)12-s + (0.632 + 0.288i)14-s + (−1.03 + 1.61i)15-s + (−0.959 + 0.281i)16-s + (−1.10 − 2.42i)18-s + (0.540 + 0.841i)20-s + (1.00 + 0.871i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.433 + 0.901i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.433 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46919 - 2.33670i\)
\(L(\frac12)\) \(\approx\) \(1.46919 - 2.33670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.926 + 1.06i)T \)
5 \( 1 + (2.03 - 0.928i)T \)
23 \( 1 + (-4.04 - 2.57i)T \)
good3 \( 1 + (-2.78 + 1.79i)T + (1.24 - 2.72i)T^{2} \)
7 \( 1 + (-0.518 - 1.76i)T + (-5.88 + 3.78i)T^{2} \)
11 \( 1 + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (0.803 - 5.58i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-12.8 - 28.1i)T^{2} \)
37 \( 1 + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-5.30 - 11.6i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-6.56 - 10.2i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-3.24 + 5.05i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (11.7 + 10.2i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (10.1 + 70.2i)T^{2} \)
73 \( 1 + (70.0 - 20.5i)T^{2} \)
79 \( 1 + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (14.9 + 6.80i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (3.88 + 6.04i)T + (-36.9 + 80.9i)T^{2} \)
97 \( 1 + (-63.5 + 73.3i)T^{2} \)
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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

−11.10231638389124599366115041670, −9.709111889906877309795467995943, −8.941317603904630412266231603796, −8.089628952087750993375657054129, −7.18449411971430131148675137645, −6.23540605216395252359231502672, −4.57094023917920176795193223485, −3.32074080681037848229978544255, −2.78692206085124573044592364420, −1.45168274110866300584840881714, 2.67898550574875172441609894601, 3.85772477111881771122765021718, 4.24707489344374600584123091736, 5.25819736451274058781282154402, 7.16001378190821093235418894815, 7.72981315436966122112188131569, 8.586521075963681136609521151283, 9.116333259545870619475105910925, 10.33567232343446395397040594388, 11.29436214158660862896612123900

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