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Properties

Label 2-1323-63.59-c1-0-31
Degree $2$
Conductor $1323$
Sign $-0.107 + 0.994i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
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  + (0.254 + 0.146i)2-s + (−0.956 − 1.65i)4-s + 3.06·5-s − 1.15i·8-s + (0.778 + 0.449i)10-s − 3.89i·11-s + (−2.02 − 1.17i)13-s + (−1.74 + 3.02i)16-s + (1.68 − 2.91i)17-s + (−2.20 + 1.27i)19-s + (−2.92 − 5.07i)20-s + (0.572 − 0.991i)22-s − 2.98i·23-s + 4.36·25-s + (−0.344 − 0.596i)26-s + ⋯
L(s)  = 1  + (0.179 + 0.103i)2-s + (−0.478 − 0.828i)4-s + 1.36·5-s − 0.406i·8-s + (0.246 + 0.142i)10-s − 1.17i·11-s + (−0.562 − 0.324i)13-s + (−0.436 + 0.755i)16-s + (0.408 − 0.706i)17-s + (−0.506 + 0.292i)19-s + (−0.654 − 1.13i)20-s + (0.122 − 0.211i)22-s − 0.621i·23-s + 0.873·25-s + (−0.0675 − 0.116i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.107 + 0.994i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (962, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.107 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.791830603\)
\(L(\frac12)\) \(\approx\) \(1.791830603\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.254 - 0.146i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 3.06T + 5T^{2} \)
11 \( 1 + 3.89iT - 11T^{2} \)
13 \( 1 + (2.02 + 1.17i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.68 + 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.20 - 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.98iT - 23T^{2} \)
29 \( 1 + (-3.67 + 2.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.409 + 0.236i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.89 + 6.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.12 - 5.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.06 - 3.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.02 - 3.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.99 - 2.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.34 + 4.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.38 - 0.800i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.787 + 1.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.6iT - 71T^{2} \)
73 \( 1 + (-0.856 - 0.494i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.63 + 8.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.49 + 9.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.15 - 3.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.98 + 2.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.413703536702654130193455174022, −8.889214695467739466980301716952, −7.86158611253497549152626676478, −6.55386121031776241926686718185, −6.01513698229137375966191786136, −5.34774345608215020882486723209, −4.54991982962482130422257555364, −3.13809095323017258200859488485, −1.99773110089917573547687836160, −0.69424051687877564012934847203, 1.75903577518784943604518063607, 2.60237509257306756288003667091, 3.83563012727908790301398224376, 4.82140645830148966370972576807, 5.48424767389114608738800797449, 6.65622442696307091851072597548, 7.30099786163432697348781542489, 8.391039430498481046188467844469, 9.061138916468704832311769937214, 9.928149858977752346470507629592

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