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Properties

Label 2-1323-63.5-c1-0-31
Degree $2$
Conductor $1323$
Sign $-0.891 + 0.453i$
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  + 1.17i·2-s + 0.611·4-s + (−2.16 − 3.75i)5-s + 3.07i·8-s + (4.42 − 2.55i)10-s + (−1.87 − 1.08i)11-s + (−2.25 − 1.30i)13-s − 2.40·16-s + (0.585 + 1.01i)17-s + (−2.09 − 1.20i)19-s + (−1.32 − 2.29i)20-s + (1.27 − 2.20i)22-s + (−3.16 + 1.82i)23-s + (−6.88 + 11.9i)25-s + (1.53 − 2.65i)26-s + ⋯
L(s)  = 1  + 0.833i·2-s + 0.305·4-s + (−0.968 − 1.67i)5-s + 1.08i·8-s + (1.39 − 0.807i)10-s + (−0.564 − 0.325i)11-s + (−0.624 − 0.360i)13-s − 0.600·16-s + (0.142 + 0.245i)17-s + (−0.480 − 0.277i)19-s + (−0.296 − 0.513i)20-s + (0.271 − 0.470i)22-s + (−0.659 + 0.380i)23-s + (−1.37 + 2.38i)25-s + (0.300 − 0.520i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.891 + 0.453i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.891 + 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09885103834\)
\(L(\frac12)\) \(\approx\) \(0.09885103834\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.17iT - 2T^{2} \)
5 \( 1 + (2.16 + 3.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.87 + 1.08i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.25 + 1.30i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.585 - 1.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.09 + 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.16 - 1.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.589 - 0.340i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.55iT - 31T^{2} \)
37 \( 1 + (-2.55 + 4.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.68 - 6.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.12 + 3.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.14T + 47T^{2} \)
53 \( 1 + (2.79 - 1.61i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.83T + 59T^{2} \)
61 \( 1 + 7.17iT - 61T^{2} \)
67 \( 1 - 6.65T + 67T^{2} \)
71 \( 1 + 1.95iT - 71T^{2} \)
73 \( 1 + (10.3 - 5.95i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 9.75T + 79T^{2} \)
83 \( 1 + (-0.796 - 1.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.04 + 5.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.36 + 1.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.904151244847419138577184947914, −8.239653913037372932893404431802, −7.83866546884249344544435993089, −6.99255918665498500811474482924, −5.82343076716819689906633194354, −5.13961082174004263476383005704, −4.46309260146820763690977577108, −3.19471097870951857002728057326, −1.66880514111227319228944821892, −0.03730158974960640506705161325, 2.11068309252571301582267114702, 2.79710955078450791936701475225, 3.65856164737747982364244681142, 4.49628665611247311832155593318, 6.11096861493916852450693858175, 6.80128555331949091113004172380, 7.49496462404673406544899521329, 8.105898113237162236805911595525, 9.610516619290761015732660901293, 10.25592407708176223851570348320

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