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Properties

Label 2-1323-63.59-c1-0-19
Degree $2$
Conductor $1323$
Sign $-0.269 - 0.963i$
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  + (2.24 + 1.29i)2-s + (2.36 + 4.09i)4-s + 1.25·5-s + 7.07i·8-s + (2.81 + 1.62i)10-s − 0.616i·11-s + (1.06 + 0.613i)13-s + (−4.44 + 7.69i)16-s + (−2.21 + 3.83i)17-s + (1.64 − 0.950i)19-s + (2.96 + 5.12i)20-s + (0.799 − 1.38i)22-s + 4.74i·23-s − 3.43·25-s + (1.59 + 2.75i)26-s + ⋯
L(s)  = 1  + (1.58 + 0.916i)2-s + (1.18 + 2.04i)4-s + 0.560·5-s + 2.50i·8-s + (0.889 + 0.513i)10-s − 0.185i·11-s + (0.294 + 0.170i)13-s + (−1.11 + 1.92i)16-s + (−0.537 + 0.930i)17-s + (0.377 − 0.218i)19-s + (0.662 + 1.14i)20-s + (0.170 − 0.295i)22-s + 0.990i·23-s − 0.686·25-s + (0.312 + 0.540i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.269 - 0.963i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.269 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.540771801\)
\(L(\frac12)\) \(\approx\) \(4.540771801\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.24 - 1.29i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 1.25T + 5T^{2} \)
11 \( 1 + 0.616iT - 11T^{2} \)
13 \( 1 + (-1.06 - 0.613i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.21 - 3.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.64 + 0.950i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.74iT - 23T^{2} \)
29 \( 1 + (-5.07 + 2.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.14 + 1.24i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.33 - 2.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.09 + 3.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.24 + 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.80 + 6.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.67 + 1.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.78 - 3.08i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.5 + 7.22i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.80 + 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (9.95 + 5.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.01 + 3.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.36 - 7.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.811 + 1.40i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.76 + 5.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.862364959070298224722316497312, −8.788252853983805894001195362390, −7.954167426146736925785547442139, −7.14666777370062026301924881894, −6.22366949698679841016071020612, −5.84210370517166641127191720355, −4.87443135086856654922944867996, −4.03587946483791050452331986018, −3.15615785338229210923556449530, −1.96786669148840725211012158214, 1.23543625994592098590051128669, 2.43262507126831980968846165471, 3.11872850887711915842268652740, 4.33994286701573446915389355079, 4.89103131550479124608734635369, 5.90568622182197058024185684273, 6.43237160148033196404375168516, 7.50084465466011227222295517286, 8.837320880496306043000314831445, 9.768240235687056647654884238188

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