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Properties

Label 2-1323-63.5-c1-0-30
Degree $2$
Conductor $1323$
Sign $-0.265 + 0.963i$
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  + 1.48i·2-s − 0.203·4-s + (−0.154 − 0.267i)5-s + 2.66i·8-s + (0.396 − 0.228i)10-s + (−2.73 − 1.58i)11-s + (−3.00 − 1.73i)13-s − 4.36·16-s + (−2.44 − 4.22i)17-s + (−4.62 − 2.67i)19-s + (0.0314 + 0.0544i)20-s + (2.34 − 4.06i)22-s + (−5.17 + 2.98i)23-s + (2.45 − 4.24i)25-s + (2.57 − 4.45i)26-s + ⋯
L(s)  = 1  + 1.04i·2-s − 0.101·4-s + (−0.0689 − 0.119i)5-s + 0.942i·8-s + (0.125 − 0.0723i)10-s + (−0.825 − 0.476i)11-s + (−0.833 − 0.481i)13-s − 1.09·16-s + (−0.592 − 1.02i)17-s + (−1.06 − 0.612i)19-s + (0.00702 + 0.0121i)20-s + (0.500 − 0.866i)22-s + (−1.07 + 0.622i)23-s + (0.490 − 0.849i)25-s + (0.504 − 0.874i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.265 + 0.963i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.265 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03240128428\)
\(L(\frac12)\) \(\approx\) \(0.03240128428\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.48iT - 2T^{2} \)
5 \( 1 + (0.154 + 0.267i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.73 + 1.58i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.00 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.44 + 4.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.62 + 2.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.17 - 2.98i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.70 - 1.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.52iT - 31T^{2} \)
37 \( 1 + (5.92 - 10.2i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.58 + 4.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.75 - 4.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.46T + 47T^{2} \)
53 \( 1 + (0.0740 - 0.0427i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.08T + 59T^{2} \)
61 \( 1 + 5.42iT - 61T^{2} \)
67 \( 1 + 0.110T + 67T^{2} \)
71 \( 1 + 7.78iT - 71T^{2} \)
73 \( 1 + (-8.32 + 4.80i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.13T + 79T^{2} \)
83 \( 1 + (4.42 + 7.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.936 - 1.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.9 + 6.34i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.113046826360907236354578193128, −8.341514211432408909173027569543, −7.74749259196647599631578179564, −6.89299004522268724255355348713, −6.25809983622318363637678415930, −5.14927407353221450016556202935, −4.76757506711428211413234387773, −3.11094674329212980066290080657, −2.17370769173196985579268830778, −0.01162126740897810802832938112, 1.94913924739913950726813745447, 2.38138098891613013867164749281, 3.79178439689480941867237981815, 4.38027910000386021510157556827, 5.68629658709180739763671160196, 6.63130790104648087468594990818, 7.44288424858141031450105159276, 8.321728307073436739337303838260, 9.369237565004717055639374893456, 10.06570712758915054188445465776

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