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Properties

Label 2-460-115.103-c1-0-11
Degree $2$
Conductor $460$
Sign $0.562 + 0.826i$
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  + (2.28 − 1.71i)3-s + (2.01 + 0.970i)5-s + (0.221 − 3.09i)7-s + (1.46 − 4.97i)9-s + (−1.36 + 2.13i)11-s + (−3.76 + 0.269i)13-s + (6.27 − 1.22i)15-s + (1.38 − 3.70i)17-s + (−2.83 + 6.20i)19-s + (−4.80 − 7.47i)21-s + (0.511 + 4.76i)23-s + (3.11 + 3.91i)25-s + (−2.18 − 5.84i)27-s + (3.12 − 1.42i)29-s + (1.18 + 8.23i)31-s + ⋯
L(s)  = 1  + (1.32 − 0.989i)3-s + (0.900 + 0.434i)5-s + (0.0837 − 1.17i)7-s + (0.486 − 1.65i)9-s + (−0.412 + 0.642i)11-s + (−1.04 + 0.0746i)13-s + (1.62 − 0.317i)15-s + (0.335 − 0.898i)17-s + (−0.650 + 1.42i)19-s + (−1.04 − 1.63i)21-s + (0.106 + 0.994i)23-s + (0.623 + 0.782i)25-s + (−0.419 − 1.12i)27-s + (0.580 − 0.265i)29-s + (0.212 + 1.47i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.562 + 0.826i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.562 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05449 - 1.08664i\)
\(L(\frac12)\) \(\approx\) \(2.05449 - 1.08664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.01 - 0.970i)T \)
23 \( 1 + (-0.511 - 4.76i)T \)
good3 \( 1 + (-2.28 + 1.71i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (-0.221 + 3.09i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (1.36 - 2.13i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (3.76 - 0.269i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (-1.38 + 3.70i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (2.83 - 6.20i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-3.12 + 1.42i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (-1.18 - 8.23i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (5.41 + 9.92i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (1.73 - 0.509i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-1.35 - 1.81i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (-6.74 + 6.74i)T - 47iT^{2} \)
53 \( 1 + (11.1 + 0.794i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (-1.04 - 0.907i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (-0.0537 + 0.00772i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (-6.18 - 1.34i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (-1.58 + 1.01i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-6.87 + 2.56i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (3.87 - 4.46i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (1.63 - 0.894i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (-0.596 + 4.15i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-12.3 - 6.72i)T + (52.4 + 81.6i)T^{2} \)
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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

−10.59134267400126642623387376356, −9.951895172990785713107471548256, −9.160863476592625510396064733708, −7.933508055101957058502921714394, −7.30001406043768442019503524537, −6.73297227308259029888473281017, −5.21645475870218558607648162027, −3.69024022230105371802162508443, −2.54985192530949898018158482613, −1.55962369132105285265355371323, 2.30623558675901775859213108311, 2.86875913372908579122394942306, 4.48493042003418623561776374090, 5.25145768625748690017832169513, 6.40497484322306286460134326383, 8.091475182366072263637427690830, 8.652851306042187026921357990176, 9.304401505811332206885974835159, 10.04152990465109030939140804001, 10.85438643389066191570907570624

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