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Properties

Label 2-1323-63.38-c1-0-18
Degree $2$
Conductor $1323$
Sign $0.965 - 0.259i$
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  + 0.122i·2-s + 1.98·4-s + (0.264 − 0.458i)5-s + 0.487i·8-s + (0.0560 + 0.0323i)10-s + (−3.64 + 2.10i)11-s + (−1.74 + 1.00i)13-s + 3.91·16-s + (2.19 − 3.79i)17-s + (4.54 − 2.62i)19-s + (0.525 − 0.910i)20-s + (−0.257 − 0.445i)22-s + (5.43 + 3.13i)23-s + (2.35 + 4.08i)25-s + (−0.123 − 0.213i)26-s + ⋯
L(s)  = 1  + 0.0865i·2-s + 0.992·4-s + (0.118 − 0.205i)5-s + 0.172i·8-s + (0.0177 + 0.0102i)10-s + (−1.09 + 0.633i)11-s + (−0.484 + 0.279i)13-s + 0.977·16-s + (0.532 − 0.921i)17-s + (1.04 − 0.601i)19-s + (0.117 − 0.203i)20-s + (−0.0548 − 0.0949i)22-s + (1.13 + 0.654i)23-s + (0.471 + 0.817i)25-s + (−0.0242 − 0.0419i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.965 - 0.259i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.170629017\)
\(L(\frac12)\) \(\approx\) \(2.170629017\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.122iT - 2T^{2} \)
5 \( 1 + (-0.264 + 0.458i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.64 - 2.10i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.74 - 1.00i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.19 + 3.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.54 + 2.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.43 - 3.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.27 - 4.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.19iT - 31T^{2} \)
37 \( 1 + (-1.61 - 2.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0994 - 0.172i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.96 + 6.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.97T + 47T^{2} \)
53 \( 1 + (3.65 + 2.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 + 6.58T + 67T^{2} \)
71 \( 1 + 8.50iT - 71T^{2} \)
73 \( 1 + (4.86 + 2.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.572T + 79T^{2} \)
83 \( 1 + (5.42 - 9.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.43 - 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.493 - 0.285i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.655511778280631279733029135335, −8.978596183704611742759640566101, −7.62465449178150355458351879185, −7.39527095688185241273153761774, −6.54000155088718055648847813174, −5.20638395874822124767892451655, −5.01368416079361819813569814761, −3.19852833729083161056080435019, −2.58269260087177942340726895758, −1.22059979324943938780222254198, 1.07409424508311127949511299237, 2.63116347572746961302269670869, 3.03943828041073112911110805161, 4.48514883685407977088261297877, 5.69423985461566623334996085060, 6.13828703268849180442335002565, 7.28652461640146989162314287913, 7.83038444009529494326242075875, 8.641019886089094179053652868829, 9.903253637297013221576465305875

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