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Properties

Label 2-1323-63.5-c1-0-6
Degree $2$
Conductor $1323$
Sign $-0.628 - 0.777i$
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  + 2.70i·2-s − 5.33·4-s + (−0.601 − 1.04i)5-s − 9.04i·8-s + (2.82 − 1.62i)10-s + (−2.15 − 1.24i)11-s + (1.63 + 0.942i)13-s + 13.8·16-s + (−0.601 − 1.04i)17-s + (6.46 + 3.73i)19-s + (3.21 + 5.56i)20-s + (3.36 − 5.83i)22-s + (2.63 − 1.52i)23-s + (1.77 − 3.07i)25-s + (−2.55 + 4.42i)26-s + ⋯
L(s)  = 1  + 1.91i·2-s − 2.66·4-s + (−0.268 − 0.465i)5-s − 3.19i·8-s + (0.892 − 0.515i)10-s + (−0.649 − 0.374i)11-s + (0.452 + 0.261i)13-s + 3.45·16-s + (−0.145 − 0.252i)17-s + (1.48 + 0.856i)19-s + (0.717 + 1.24i)20-s + (0.718 − 1.24i)22-s + (0.549 − 0.317i)23-s + (0.355 − 0.615i)25-s + (−0.500 + 0.867i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.628 - 0.777i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.628 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.208283904\)
\(L(\frac12)\) \(\approx\) \(1.208283904\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70iT - 2T^{2} \)
5 \( 1 + (0.601 + 1.04i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.15 + 1.24i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.63 - 0.942i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.601 + 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.46 - 3.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.63 + 1.52i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.173 + 0.100i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.50iT - 31T^{2} \)
37 \( 1 + (0.865 - 1.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.36 - 5.82i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.00656 - 0.0113i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.43T + 47T^{2} \)
53 \( 1 + (-8.58 + 4.95i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 11.2iT - 61T^{2} \)
67 \( 1 + 5.15T + 67T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + (-7.51 + 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.49T + 79T^{2} \)
83 \( 1 + (1.60 + 2.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.98 - 6.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.06 + 1.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.604764225740140019689800184367, −8.567485296202136559380688107706, −8.360546384208959149322049652943, −7.39072330694166713862318205765, −6.75596880282336715370190418346, −5.75550565342263695996411064499, −5.17614030489735189395324564834, −4.34429731574114625854179927293, −3.27663283568203634790365038480, −0.869540799006626525937010832217, 0.77853394364952245572372416041, 2.11336963798105390706939984877, 3.08042190134435712251642577405, 3.70228898848335503230862552151, 4.85922543751953212747724151081, 5.51342804670186282643303631433, 7.09475503556444332991550257506, 7.972607631318237999199714006416, 8.949378566507984302736206111157, 9.543512481027391579450533245015

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