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Properties

Label 2-1323-63.41-c1-0-16
Degree $2$
Conductor $1323$
Sign $0.983 + 0.183i$
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  + (−1.58 + 0.916i)2-s + (0.678 − 1.17i)4-s + (0.322 − 0.559i)5-s − 1.17i·8-s + 1.18i·10-s + (−4.60 + 2.65i)11-s + (−4.44 − 2.56i)13-s + (2.43 + 4.22i)16-s + 1.62·17-s + 2.41i·19-s + (−0.437 − 0.758i)20-s + (4.86 − 8.43i)22-s + (1.27 + 0.735i)23-s + (2.29 + 3.96i)25-s + 9.39·26-s + ⋯
L(s)  = 1  + (−1.12 + 0.647i)2-s + (0.339 − 0.587i)4-s + (0.144 − 0.250i)5-s − 0.416i·8-s + 0.374i·10-s + (−1.38 + 0.801i)11-s + (−1.23 − 0.711i)13-s + (0.609 + 1.05i)16-s + 0.395·17-s + 0.553i·19-s + (−0.0978 − 0.169i)20-s + (1.03 − 1.79i)22-s + (0.265 + 0.153i)23-s + (0.458 + 0.793i)25-s + 1.84·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.983 + 0.183i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.983 + 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6098269833\)
\(L(\frac12)\) \(\approx\) \(0.6098269833\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 0.916i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.322 + 0.559i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.60 - 2.65i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.44 + 2.56i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.62T + 17T^{2} \)
19 \( 1 - 2.41iT - 19T^{2} \)
23 \( 1 + (-1.27 - 0.735i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.43 + 3.71i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.90 - 2.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.99T + 37T^{2} \)
41 \( 1 + (-5.99 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.51 + 2.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.54 + 2.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.36iT - 53T^{2} \)
59 \( 1 + (-1.47 + 2.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.18 + 5.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.07 + 8.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 + (3.48 + 6.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.51 - 6.09i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.32T + 89T^{2} \)
97 \( 1 + (-14.3 + 8.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.598512179031090574381790952679, −8.694450017063880424034459102832, −7.900044378499634959775407308041, −7.46070806524454001815552778055, −6.64958525979947106328490672399, −5.42636347423735645723760221449, −4.83876568670777933809834041996, −3.34599042523272787205995589749, −2.10918109462924437953356259583, −0.48655863320543528757246911804, 0.900211413755365850236397441294, 2.49078761245780681162817705585, 2.86382387871700490478525819784, 4.64849660512220329586730497791, 5.36703121298347100912831669280, 6.56175954511746938528687131494, 7.49507977465264363468954073300, 8.285007191654941340733935306397, 8.855578246736534603386100479291, 9.961935580700577103192847940450

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