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Properties

Label 2-460-23.4-c1-0-7
Degree $2$
Conductor $460$
Sign $0.0309 + 0.999i$
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  + (1.36 − 0.400i)3-s + (−0.841 − 0.540i)5-s + (−0.649 − 4.51i)7-s + (−0.827 + 0.531i)9-s + (1.47 − 3.22i)11-s + (−0.294 + 2.05i)13-s + (−1.36 − 0.400i)15-s + (−2.58 − 2.97i)17-s + (0.343 − 0.396i)19-s + (−2.69 − 5.89i)21-s + (1.09 − 4.66i)23-s + (0.415 + 0.909i)25-s + (−3.70 + 4.27i)27-s + (3.44 + 3.97i)29-s + (8.46 + 2.48i)31-s + ⋯
L(s)  = 1  + (0.786 − 0.230i)3-s + (−0.376 − 0.241i)5-s + (−0.245 − 1.70i)7-s + (−0.275 + 0.177i)9-s + (0.444 − 0.973i)11-s + (−0.0818 + 0.568i)13-s + (−0.351 − 0.103i)15-s + (−0.625 − 0.722i)17-s + (0.0787 − 0.0909i)19-s + (−0.587 − 1.28i)21-s + (0.227 − 0.973i)23-s + (0.0830 + 0.181i)25-s + (−0.712 + 0.822i)27-s + (0.639 + 0.737i)29-s + (1.52 + 0.446i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0309 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0309 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.0309 + 0.999i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.0309 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07660 - 1.04382i\)
\(L(\frac12)\) \(\approx\) \(1.07660 - 1.04382i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-1.09 + 4.66i)T \)
good3 \( 1 + (-1.36 + 0.400i)T + (2.52 - 1.62i)T^{2} \)
7 \( 1 + (0.649 + 4.51i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-1.47 + 3.22i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.294 - 2.05i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (2.58 + 2.97i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-0.343 + 0.396i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-3.44 - 3.97i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-8.46 - 2.48i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-6.43 + 4.13i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-1.26 - 0.810i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (11.0 - 3.23i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + (0.374 + 2.60i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.598 + 4.16i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-4.58 - 1.34i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-5.62 - 12.3i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (4.10 + 8.98i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-4.44 + 5.12i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (2.29 - 15.9i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-3.79 + 2.44i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (11.6 - 3.42i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (0.311 + 0.200i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.92554263680768913574552296049, −9.904293079102771481277685884560, −8.832924220690983629151421303338, −8.205656211019897469881088535270, −7.19056376040712250765260875272, −6.49514966455000766247897777203, −4.79137292673668980475812339830, −3.85621210509351173659610995284, −2.78966802444882035881902502296, −0.875396366283111832903821799492, 2.25257405400406207186033319973, 3.11009063645792860956869229744, 4.36234235973500080808336145890, 5.69496067379292938949979570127, 6.56761894717628485373422844013, 7.944843687718061694485394860581, 8.586384805630876854252759758614, 9.407167055047157392182994391678, 10.09285177696368996847817356692, 11.56925050453968155269446391793

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