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Properties

Label 2-1323-63.4-c1-0-25
Degree $2$
Conductor $1323$
Sign $-0.394 + 0.919i$
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  + (0.119 − 0.207i)2-s + (0.971 + 1.68i)4-s − 2.59·5-s + 0.942·8-s + (−0.309 + 0.536i)10-s − 4.18·11-s + (1.84 − 3.18i)13-s + (−1.83 + 3.16i)16-s + (−0.855 + 1.48i)17-s + (−3.57 − 6.19i)19-s + (−2.51 − 4.36i)20-s + (−0.499 + 0.866i)22-s + 5.12·23-s + 1.71·25-s + (−0.440 − 0.762i)26-s + ⋯
L(s)  = 1  + (0.0845 − 0.146i)2-s + (0.485 + 0.841i)4-s − 1.15·5-s + 0.333·8-s + (−0.0979 + 0.169i)10-s − 1.26·11-s + (0.510 − 0.884i)13-s + (−0.457 + 0.792i)16-s + (−0.207 + 0.359i)17-s + (−0.820 − 1.42i)19-s + (−0.562 − 0.975i)20-s + (−0.106 + 0.184i)22-s + 1.06·23-s + 0.343·25-s + (−0.0863 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.394 + 0.919i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.394 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6300207871\)
\(L(\frac12)\) \(\approx\) \(0.6300207871\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.119 + 0.207i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 2.59T + 5T^{2} \)
11 \( 1 + 4.18T + 11T^{2} \)
13 \( 1 + (-1.84 + 3.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.855 - 1.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.57 + 6.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.12T + 23T^{2} \)
29 \( 1 + (1.06 + 1.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.26 + 5.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.830 + 1.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.10 + 8.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.830 - 1.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.66 - 8.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.32 + 9.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.03 + 5.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.99 - 6.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.13 + 7.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.23T + 71T^{2} \)
73 \( 1 + (-3.57 + 6.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.91 + 8.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.44 + 5.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.51 - 4.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.53 - 2.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.150065430868142034447077643596, −8.362548632990767840844350351463, −7.72077703326448254691485903307, −7.24316103982906494868187146932, −6.14630591075356224771046973567, −4.95278986775004596653024409043, −4.05344727120241779509431490356, −3.18620812127474009745409839222, −2.34497216771989950401843735928, −0.24694211979477312987432548151, 1.45638642423796687837297759581, 2.76250764390182906548691438075, 3.94632332596423037850733178734, 4.85316969164427442648893241808, 5.71170144064586963726700370383, 6.68925082504889708089090757475, 7.38480578709509466932790866750, 8.162855045421543358798646386337, 8.994070779633626096097396897327, 10.05616535369409940652821597418

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