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Properties

Label 2-1323-63.59-c1-0-4
Degree $2$
Conductor $1323$
Sign $0.979 - 0.199i$
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  + (−2.05 − 1.18i)2-s + (1.81 + 3.14i)4-s − 3.43·5-s − 3.86i·8-s + (7.05 + 4.07i)10-s − 0.313i·11-s + (−5.09 − 2.94i)13-s + (−0.958 + 1.65i)16-s + (0.476 − 0.825i)17-s + (−1.09 + 0.630i)19-s + (−6.23 − 10.7i)20-s + (−0.372 + 0.645i)22-s − 6.82i·23-s + 6.80·25-s + (6.98 + 12.0i)26-s + ⋯
L(s)  = 1  + (−1.45 − 0.838i)2-s + (0.907 + 1.57i)4-s − 1.53·5-s − 1.36i·8-s + (2.23 + 1.28i)10-s − 0.0946i·11-s + (−1.41 − 0.816i)13-s + (−0.239 + 0.414i)16-s + (0.115 − 0.200i)17-s + (−0.250 + 0.144i)19-s + (−1.39 − 2.41i)20-s + (−0.0794 + 0.137i)22-s − 1.42i·23-s + 1.36·25-s + (1.36 + 2.37i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.979 - 0.199i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.979 - 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2271751677\)
\(L(\frac12)\) \(\approx\) \(0.2271751677\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 + 1.18i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 3.43T + 5T^{2} \)
11 \( 1 + 0.313iT - 11T^{2} \)
13 \( 1 + (5.09 + 2.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.476 + 0.825i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 - 0.630i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.82iT - 23T^{2} \)
29 \( 1 + (3.43 - 1.98i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.53 - 2.61i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.68 + 4.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0699 - 0.121i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.44 - 2.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.00 + 1.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.3 - 5.98i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.824 + 1.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.57 + 1.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.934 - 1.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (0.354 + 0.204i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.23 - 9.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.00 - 6.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.05 + 1.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.5 + 6.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.694411441942519047786666562103, −8.787563637548211470159270321116, −8.228208126534677316317110598182, −7.45753326636897297667079454313, −7.05150021223714849513411877181, −5.35161732049463603810015243592, −4.21295148982001905137252268442, −3.21176836532894740255638492395, −2.32707983596550185953352785545, −0.66534864241143390718086341167, 0.26122224468840269767490905146, 1.90334383035815955151241451034, 3.54728929991755132143312966751, 4.55417483780794122468866045773, 5.67529042179466390370245379692, 6.86756957852575170619015750299, 7.41263693507410336150300423615, 7.81311332246943067189967131447, 8.732293228540209261798380182362, 9.387234148124552249498137369342

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