Properties

Label 2-1323-63.25-c1-0-25
Degree $2$
Conductor $1323$
Sign $0.933 - 0.359i$
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  + 2.71·2-s + 5.37·4-s + (−0.793 + 1.37i)5-s + 9.15·8-s + (−2.15 + 3.73i)10-s + (−0.674 − 1.16i)11-s + (1.58 + 2.75i)13-s + 14.1·16-s + (1.40 − 2.42i)17-s + (−0.312 − 0.541i)19-s + (−4.26 + 7.38i)20-s + (−1.83 − 3.17i)22-s + (−0.142 + 0.246i)23-s + (1.24 + 2.15i)25-s + (4.31 + 7.47i)26-s + ⋯
L(s)  = 1  + 1.91·2-s + 2.68·4-s + (−0.354 + 0.614i)5-s + 3.23·8-s + (−0.681 + 1.17i)10-s + (−0.203 − 0.352i)11-s + (0.440 + 0.763i)13-s + 3.52·16-s + (0.339 − 0.588i)17-s + (−0.0717 − 0.124i)19-s + (−0.952 + 1.65i)20-s + (−0.390 − 0.676i)22-s + (−0.0296 + 0.0514i)23-s + (0.248 + 0.430i)25-s + (0.846 + 1.46i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.933 - 0.359i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.933 - 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(5.413632507\)
\(L(\frac12)\) \(\approx\) \(5.413632507\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.71T + 2T^{2} \)
5 \( 1 + (0.793 - 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.674 + 1.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.58 - 2.75i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.40 + 2.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.312 + 0.541i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.142 - 0.246i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.27 - 3.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.43T + 31T^{2} \)
37 \( 1 + (4.01 + 6.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.01 + 8.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.12 - 5.42i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + (-1.39 + 2.41i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.57T + 59T^{2} \)
61 \( 1 + 0.385T + 61T^{2} \)
67 \( 1 + 2.53T + 67T^{2} \)
71 \( 1 - 1.45T + 71T^{2} \)
73 \( 1 + (-0.234 + 0.405i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + (-6.99 + 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.29 + 2.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.22 + 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.06001285125834674934927741816, −8.724133619094491420270373314651, −7.55706724943286174548942689571, −6.94361420938292663088405470394, −6.25225746847995865760155882885, −5.34165296277938146626341796130, −4.56578363571083494107359997692, −3.55103531230910929374242361563, −3.01377821699918058107472484657, −1.78064299641430404417096867330, 1.46978050315112993906775463550, 2.79400896479495176889038590603, 3.64285584160303480059317516075, 4.56270438474160822324911113309, 5.13453953893549705134623390215, 6.08755640165641020234276855664, 6.71628917331921606613506795055, 7.88328174599132213049814622916, 8.343472250911054594536866299073, 9.986265874333401967713952993914

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