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Properties

Label 2-1323-63.41-c1-0-17
Degree $2$
Conductor $1323$
Sign $0.958 - 0.284i$
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 + (1.53 − 2.65i)5-s − 1.15i·8-s + 0.899i·10-s + (−3.37 + 1.94i)11-s + (2.02 + 1.17i)13-s + (−1.74 − 3.02i)16-s + 3.36·17-s + 2.54i·19-s + (2.92 + 5.07i)20-s + (0.572 − 0.991i)22-s + (2.58 + 1.49i)23-s + (−2.18 − 3.78i)25-s − 0.688·26-s + ⋯
L(s)  = 1  + (−0.179 + 0.103i)2-s + (−0.478 + 0.828i)4-s + (0.684 − 1.18i)5-s − 0.406i·8-s + 0.284i·10-s + (−1.01 + 0.587i)11-s + (0.562 + 0.324i)13-s + (−0.436 − 0.755i)16-s + 0.816·17-s + 0.584i·19-s + (0.654 + 1.13i)20-s + (0.122 − 0.211i)22-s + (0.538 + 0.310i)23-s + (−0.436 − 0.756i)25-s − 0.135·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.958 - 0.284i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.958 - 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.483485222\)
\(L(\frac12)\) \(\approx\) \(1.483485222\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.53 + 2.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.37 - 1.94i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.02 - 1.17i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.36T + 17T^{2} \)
19 \( 1 - 2.54iT - 19T^{2} \)
23 \( 1 + (-2.58 - 1.49i)T + (11.5 + 19.9i)T^{2} \)
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 \( 1 - 7.78T + 37T^{2} \)
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 \( 1 + 5.76iT - 53T^{2} \)
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 \( 1 - 0.988iT - 73T^{2} \)
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 \( 1 - 4.31T + 89T^{2} \)
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

−9.524614220635514159572609204461, −8.887006919345673878107557192313, −8.054586978652949779627813617335, −7.57355656972951025705602577489, −6.30601070266534880518626687612, −5.30994807473597796450546934680, −4.67609795653319080158227689140, −3.67196394607414099982085141599, −2.40654881619726505029396636606, −0.988026386784971438632065320939, 0.908023778330807532956245976056, 2.42437997709071025118848219595, 3.17748401378071107851027545841, 4.62241646000164685117140481466, 5.64680033199869757218958050335, 6.07909986500521639050356626654, 7.05646458660047112036693363504, 8.064621695830360476959468556416, 8.904657497942977125810838888019, 9.781529735855059385167633617564

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