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Properties

Label 2-1323-147.53-c0-0-0
Degree $2$
Conductor $1323$
Sign $-0.304 - 0.952i$
Analytic cond. $0.660263$
Root an. cond. $0.812565$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.365 + 0.930i)4-s + (−0.900 + 0.433i)7-s + (−1.72 + 0.829i)13-s + (−0.733 + 0.680i)16-s + (0.900 + 1.56i)19-s + (0.826 − 0.563i)25-s + (−0.733 − 0.680i)28-s + (−0.365 + 0.632i)31-s + (0.266 − 0.680i)37-s + (−0.0332 − 0.145i)43-s + (0.623 − 0.781i)49-s + (−1.40 − 1.29i)52-s + (−0.722 + 1.84i)61-s + (−0.900 − 0.433i)64-s + (0.733 − 1.26i)67-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)4-s + (−0.900 + 0.433i)7-s + (−1.72 + 0.829i)13-s + (−0.733 + 0.680i)16-s + (0.900 + 1.56i)19-s + (0.826 − 0.563i)25-s + (−0.733 − 0.680i)28-s + (−0.365 + 0.632i)31-s + (0.266 − 0.680i)37-s + (−0.0332 − 0.145i)43-s + (0.623 − 0.781i)49-s + (−1.40 − 1.29i)52-s + (−0.722 + 1.84i)61-s + (−0.900 − 0.433i)64-s + (0.733 − 1.26i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.304 - 0.952i$
Analytic conductor: \(0.660263\)
Root analytic conductor: \(0.812565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ -0.304 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8744164264\)
\(L(\frac12)\) \(\approx\) \(0.8744164264\)
\(L(1)\) 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 \( 1 + (0.900 - 0.433i)T \)
good2 \( 1 + (-0.365 - 0.930i)T^{2} \)
5 \( 1 + (-0.826 + 0.563i)T^{2} \)
11 \( 1 + (0.988 + 0.149i)T^{2} \)
13 \( 1 + (1.72 - 0.829i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.955 - 0.294i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.266 + 0.680i)T + (-0.733 - 0.680i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.0332 + 0.145i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.365 - 0.930i)T^{2} \)
53 \( 1 + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (-0.826 - 0.563i)T^{2} \)
61 \( 1 + (0.722 - 1.84i)T + (-0.733 - 0.680i)T^{2} \)
67 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-1.03 + 0.702i)T + (0.365 - 0.930i)T^{2} \)
79 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
97 \( 1 - 0.149T + T^{2} \)
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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

−9.893048913371108208135267522300, −9.323127620398998621707187307451, −8.437200130750189034234587309915, −7.49566187715192005890732919393, −6.96540735883124229271029826363, −6.06819587006254402618703511753, −4.98208926594141016309229032458, −3.88899538134162519729992030138, −2.99484319327926516948411940087, −2.10627886969388468673001555287, 0.70382878742381749896769433956, 2.42159256242044056290157225166, 3.21132477476918399903424176351, 4.76945830215064508469700892100, 5.28419879812329132128895633175, 6.38483748453227092188949672211, 7.09095795580291583554858695181, 7.67490558481736105701232898845, 9.162489240642653588779069164025, 9.634526005608218515105016002988

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