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Properties

Label 2-1323-9.7-c1-0-14
Degree $2$
Conductor $1323$
Sign $0.989 - 0.145i$
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  + (0.849 − 1.47i)2-s + (−0.444 − 0.769i)4-s + (1.79 + 3.10i)5-s + 1.88·8-s + 6.09·10-s + (−1.40 + 2.43i)11-s + (0.5 + 0.866i)13-s + (2.49 − 4.31i)16-s − 4.11·17-s + 0.888·19-s + (1.59 − 2.76i)20-s + (2.38 + 4.13i)22-s + (2.93 + 5.08i)23-s + (−3.93 + 6.82i)25-s + 1.69·26-s + ⋯
L(s)  = 1  + (0.600 − 1.04i)2-s + (−0.222 − 0.384i)4-s + (0.802 + 1.38i)5-s + 0.667·8-s + 1.92·10-s + (−0.423 + 0.733i)11-s + (0.138 + 0.240i)13-s + (0.623 − 1.07i)16-s − 0.997·17-s + 0.203·19-s + (0.356 − 0.617i)20-s + (0.509 + 0.882i)22-s + (0.612 + 1.06i)23-s + (−0.787 + 1.36i)25-s + 0.333·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.989 - 0.145i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.989 - 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.733915035\)
\(L(\frac12)\) \(\approx\) \(2.733915035\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.849 + 1.47i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.79 - 3.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.40 - 2.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.11T + 17T^{2} \)
19 \( 1 - 0.888T + 19T^{2} \)
23 \( 1 + (-2.93 - 5.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.849 - 1.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.49 + 6.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.76T + 37T^{2} \)
41 \( 1 + (-2.70 - 4.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.60 - 4.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.33 + 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.123T + 53T^{2} \)
59 \( 1 + (-4.43 - 7.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.93 + 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.15 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.87T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + (-3.54 + 6.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.05 + 3.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.60T + 89T^{2} \)
97 \( 1 + (-3.66 + 6.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.823481960215475884213411397038, −9.336345103023807331900836891114, −7.79568996140009908928008632609, −7.14782728326546765930230335525, −6.32783263024313804384296774459, −5.30064130726495520714260941970, −4.29560433734837518888377513956, −3.28671302026344947800047341609, −2.49022990805779582779462559220, −1.76617313827421943088596525522, 0.959147395340411066769004114683, 2.30016943899736280641930683705, 3.94056782090519340924507797513, 4.93638861308477700226044222074, 5.35631106018319872019443444613, 6.16054487269092100459347499199, 6.92030698961121385745287050834, 8.035844792324329767183263503597, 8.658859188276740665480286185771, 9.317607950618066358470961629619

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