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Properties

Label 2-1323-63.5-c1-0-0
Degree $2$
Conductor $1323$
Sign $0.999 + 0.00507i$
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  − 2.57i·2-s − 4.64·4-s + (−1.16 − 2.01i)5-s + 6.82i·8-s + (−5.20 + 3.00i)10-s + (−3.78 − 2.18i)11-s + (−1.14 − 0.660i)13-s + 8.30·16-s + (2.89 + 5.01i)17-s + (0.584 + 0.337i)19-s + (5.41 + 9.38i)20-s + (−5.62 + 9.74i)22-s + (−4.81 + 2.78i)23-s + (−0.218 + 0.379i)25-s + (−1.70 + 2.94i)26-s + ⋯
L(s)  = 1  − 1.82i·2-s − 2.32·4-s + (−0.521 − 0.903i)5-s + 2.41i·8-s + (−1.64 + 0.950i)10-s + (−1.14 − 0.658i)11-s + (−0.317 − 0.183i)13-s + 2.07·16-s + (0.701 + 1.21i)17-s + (0.134 + 0.0774i)19-s + (1.21 + 2.09i)20-s + (−1.19 + 2.07i)22-s + (−1.00 + 0.580i)23-s + (−0.0437 + 0.0758i)25-s + (−0.333 + 0.578i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.999 + 0.00507i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.999 + 0.00507i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1029847306\)
\(L(\frac12)\) \(\approx\) \(0.1029847306\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.57iT - 2T^{2} \)
5 \( 1 + (1.16 + 2.01i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.78 + 2.18i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.14 + 0.660i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.89 - 5.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.584 - 0.337i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.81 - 2.78i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.86 - 2.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.01iT - 31T^{2} \)
37 \( 1 + (1.50 - 2.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.29 + 5.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.89 - 6.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.493T + 47T^{2} \)
53 \( 1 + (3.59 - 2.07i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.31T + 59T^{2} \)
61 \( 1 - 2.05iT - 61T^{2} \)
67 \( 1 + 4.82T + 67T^{2} \)
71 \( 1 + 1.17iT - 71T^{2} \)
73 \( 1 + (-13.0 + 7.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + (-5.32 - 9.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.66 + 2.87i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.7 - 7.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.839155999875145217624582577096, −9.052123417538829451185503627082, −8.177484275648267429898180276187, −7.79320847075425486248196551137, −5.84647109961816732039698361985, −5.12349077909574340853142827649, −4.14937344152282726934850760693, −3.45516138484736930271434819344, −2.36496122970621385537704327604, −1.17986233067790661265567841955, 0.04792640048897154137774474759, 2.63755151396814309567928482752, 3.87946999313388682239926959060, 4.88846226744724750932769697420, 5.53984625696769743625764145494, 6.54974695366447726944968796271, 7.38522025600414079110055142006, 7.55417333592012178775400485245, 8.450408500198899196595061762600, 9.499245853669630224349119493447

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