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Properties

Label 2-1323-63.5-c1-0-3
Degree $2$
Conductor $1323$
Sign $0.00194 + 0.999i$
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  + 2.37i·2-s − 3.62·4-s + (1.71 + 2.97i)5-s − 3.86i·8-s + (−7.05 + 4.07i)10-s + (0.271 + 0.156i)11-s + (−5.09 − 2.94i)13-s + 1.91·16-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 + (−5.91 + 3.41i)23-s + (−3.40 + 5.89i)25-s + (6.98 − 12.0i)26-s + ⋯
L(s)  = 1  + 1.67i·2-s − 1.81·4-s + (0.768 + 1.33i)5-s − 1.36i·8-s + (−2.23 + 1.28i)10-s + (0.0819 + 0.0473i)11-s + (−1.41 − 0.816i)13-s + 0.479·16-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 + (−1.23 + 0.711i)23-s + (−0.680 + 1.17i)25-s + (1.36 − 2.37i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00194 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00194 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.00194 + 0.999i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.00194 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9072943064\)
\(L(\frac12)\) \(\approx\) \(0.9072943064\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.37iT - 2T^{2} \)
5 \( 1 + (-1.71 - 2.97i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.271 - 0.156i)T + (5.5 + 9.52i)T^{2} \)
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 \( 1 + (5.91 - 3.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.43 - 1.98i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.23iT - 31T^{2} \)
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 \( 1 + 2.01T + 47T^{2} \)
53 \( 1 + (10.3 - 5.98i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.64T + 59T^{2} \)
61 \( 1 - 2.97iT - 61T^{2} \)
67 \( 1 + 1.86T + 67T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (-0.354 + 0.204i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
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

−9.814327684165726835999274520137, −9.560764436260343963040005287263, −8.228666644544093966856459368593, −7.53776263562768447078943621019, −7.04374878498924317121259364311, −6.09342871292690102568537134893, −5.67363315280928692271754427399, −4.68564599189202720170749946724, −3.36935189069255094135506051454, −2.20987991322456306229480141004, 0.35280255260408070737659138442, 1.71870747525532527305131771142, 2.30427463012239666335539391345, 3.66399943025353574537048539624, 4.70569396970748343605682534653, 5.09496699602926349517289098947, 6.35963079815806379005827760377, 7.66071952135472925935563934974, 8.768596797634622650339134907292, 9.293352954220785287606969185584

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