Properties

Label 2-465-465.119-c1-0-28
Degree $2$
Conductor $465$
Sign $0.786 - 0.617i$
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  − 1.73·2-s + (1.39 + 1.03i)3-s + 1.02·4-s + (2.06 − 0.847i)5-s + (−2.41 − 1.79i)6-s + (3.83 + 2.21i)7-s + 1.69·8-s + (0.873 + 2.86i)9-s + (−3.59 + 1.47i)10-s + (−1.12 − 1.94i)11-s + (1.42 + 1.05i)12-s + (1.39 + 2.41i)13-s + (−6.67 − 3.85i)14-s + (3.75 + 0.953i)15-s − 4.99·16-s + (−4.36 − 2.52i)17-s + ⋯
L(s)  = 1  − 1.22·2-s + (0.803 + 0.595i)3-s + 0.511·4-s + (0.925 − 0.379i)5-s + (−0.987 − 0.731i)6-s + (1.45 + 0.837i)7-s + 0.600·8-s + (0.291 + 0.956i)9-s + (−1.13 + 0.466i)10-s + (−0.338 − 0.586i)11-s + (0.410 + 0.304i)12-s + (0.386 + 0.668i)13-s + (−1.78 − 1.02i)14-s + (0.969 + 0.246i)15-s − 1.24·16-s + (−1.05 − 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.786 - 0.617i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.786 - 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20521 + 0.416463i\)
\(L(\frac12)\) \(\approx\) \(1.20521 + 0.416463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.39 - 1.03i)T \)
5 \( 1 + (-2.06 + 0.847i)T \)
31 \( 1 + (5.53 - 0.622i)T \)
good2 \( 1 + 1.73T + 2T^{2} \)
7 \( 1 + (-3.83 - 2.21i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.12 + 1.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.39 - 2.41i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.36 + 2.52i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.25 + 3.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.41iT - 23T^{2} \)
29 \( 1 - 3.29T + 29T^{2} \)
37 \( 1 + (3.79 - 6.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.237 - 0.137i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.74 - 4.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.70T + 47T^{2} \)
53 \( 1 + (0.556 - 0.321i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.97 - 5.75i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 10.8iT - 61T^{2} \)
67 \( 1 + (-11.4 + 6.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.93 - 4.00i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.65 + 4.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.53 - 2.61i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.1 - 7.01i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.41T + 89T^{2} \)
97 \( 1 + 17.5iT - 97T^{2} \)
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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

−10.95653878242646200894382078617, −9.931974521900029927646787558011, −9.099509496867883622448456892162, −8.654528862298215787534454576732, −8.124060534056068259152213342836, −6.80217625446457924919356720368, −5.15470792120686145028097939890, −4.62926612783197337775892942057, −2.55198203134753802888298391864, −1.61245486580951711466839080137, 1.37544935359475896027681626815, 2.08955034145921118021789860406, 3.92025481530692870406507850640, 5.33959439508614922523031524036, 6.85874101996239564222570116400, 7.55899966395115715643232210964, 8.252136288750495514262893831017, 9.005956617164997531033424726643, 10.05409055564019457315769428094, 10.55141271835561073106568711665

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