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Properties

Label 2-1323-63.38-c1-0-31
Degree $2$
Conductor $1323$
Sign $-0.135 + 0.990i$
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  − 0.641i·2-s + 1.58·4-s + (1.10 − 1.91i)5-s − 2.30i·8-s + (−1.22 − 0.709i)10-s + (−2.93 + 1.69i)11-s + (1.56 − 0.901i)13-s + 1.69·16-s + (2.98 − 5.16i)17-s + (−1.42 + 0.822i)19-s + (1.75 − 3.04i)20-s + (1.08 + 1.88i)22-s + (2.05 + 1.18i)23-s + (0.0556 + 0.0963i)25-s + (−0.578 − 1.00i)26-s + ⋯
L(s)  = 1  − 0.453i·2-s + 0.794·4-s + (0.494 − 0.856i)5-s − 0.813i·8-s + (−0.388 − 0.224i)10-s + (−0.885 + 0.511i)11-s + (0.432 − 0.249i)13-s + 0.424·16-s + (0.723 − 1.25i)17-s + (−0.326 + 0.188i)19-s + (0.392 − 0.680i)20-s + (0.232 + 0.401i)22-s + (0.428 + 0.247i)23-s + (0.0111 + 0.0192i)25-s + (−0.113 − 0.196i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.135 + 0.990i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.135 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.168195406\)
\(L(\frac12)\) \(\approx\) \(2.168195406\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.641iT - 2T^{2} \)
5 \( 1 + (-1.10 + 1.91i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.93 - 1.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.56 + 0.901i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.98 + 5.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.42 - 0.822i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.05 - 1.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.44 + 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.7iT - 31T^{2} \)
37 \( 1 + (0.849 + 1.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.455 + 0.788i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.96 - 3.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.246T + 47T^{2} \)
53 \( 1 + (-6.82 - 3.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 - 1.41iT - 61T^{2} \)
67 \( 1 + 7.98T + 67T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + (0.369 + 0.213i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.98T + 79T^{2} \)
83 \( 1 + (4.28 - 7.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.26 + 9.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.30 - 3.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.716503690790463453466025278120, −8.692876925386941167256231211189, −7.68294361720603596099201651274, −7.11962701963147773161564822980, −5.86619411919226915630518374155, −5.34540214195267450048711965670, −4.19909956092465224820732838943, −2.97640686127383837819539289544, −2.08747227127097328970854628946, −0.897393565785493010164065829466, 1.65877702566535744034035941276, 2.73931951719179234687341977793, 3.53027150631914112100723248681, 5.12164206661552406792970733924, 5.88876712543491429899993485215, 6.57897491166597857622740877720, 7.21558846481777083403172154392, 8.201802260051877815632912350616, 8.764652197620716156535842577795, 10.27471831092705607872555096117

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