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Properties

Label 2-1323-63.5-c1-0-34
Degree $2$
Conductor $1323$
Sign $-0.235 - 0.971i$
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  − 1.73i·2-s − 0.999·4-s + (−1.5 − 2.59i)5-s − 1.73i·8-s + (−4.5 + 2.59i)10-s + (−1.5 − 0.866i)11-s + (−1.5 − 0.866i)13-s − 5·16-s + (−1.5 − 2.59i)17-s + (4.5 + 2.59i)19-s + (1.49 + 2.59i)20-s + (−1.49 + 2.59i)22-s + (−4.5 + 2.59i)23-s + (−2 + 3.46i)25-s + (−1.49 + 2.59i)26-s + ⋯
L(s)  = 1  − 1.22i·2-s − 0.499·4-s + (−0.670 − 1.16i)5-s − 0.612i·8-s + (−1.42 + 0.821i)10-s + (−0.452 − 0.261i)11-s + (−0.416 − 0.240i)13-s − 1.25·16-s + (−0.363 − 0.630i)17-s + (1.03 + 0.596i)19-s + (0.335 + 0.580i)20-s + (−0.319 + 0.553i)22-s + (−0.938 + 0.541i)23-s + (−0.400 + 0.692i)25-s + (−0.294 + 0.509i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.235 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7710555876\)
\(L(\frac12)\) \(\approx\) \(0.7710555876\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT - 2T^{2} \)
5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.5 - 2.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 2.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (7.5 - 4.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (-4.5 + 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.430952200025500711465995758466, −8.241252654712571118492266540321, −7.77844876444375770307166673329, −6.59133049012727989235319212468, −5.34135586073792486108463949430, −4.57652760072173003380963784578, −3.66969626122684306767588749446, −2.72755536263351577810185112601, −1.45511418391115584070295651270, −0.31432228350817491950959462870, 2.24223482028349407986479729627, 3.25825189226011746353617589519, 4.46357951635280739302406637157, 5.36953059002454430937301952311, 6.38185004607680493565648384153, 6.99129590201556492953195970126, 7.54022840388427810196141323217, 8.261749417963911313119269930079, 9.142918382446096905217017527247, 10.30807249642145749881106989024

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