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Properties

Label 2-363-33.8-c1-0-17
Degree $2$
Conductor $363$
Sign $0.622 + 0.782i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
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  + (−0.809 + 0.587i)2-s + (−1.03 + 1.38i)3-s + (−0.309 + 0.951i)4-s + (1.66 − 2.28i)5-s + (0.0222 − 1.73i)6-s + (−1.34 − 0.437i)7-s + (−0.927 − 2.85i)8-s + (−0.853 − 2.87i)9-s + 2.82i·10-s + (−0.999 − 1.41i)12-s + (−2.49 − 3.43i)13-s + (1.34 − 0.437i)14-s + (1.45 + 4.67i)15-s + (0.809 + 0.587i)16-s + (−4.85 − 3.52i)17-s + (2.38 + 1.82i)18-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.598 + 0.801i)3-s + (−0.154 + 0.475i)4-s + (0.743 − 1.02i)5-s + (0.00907 − 0.707i)6-s + (−0.508 − 0.165i)7-s + (−0.327 − 1.00i)8-s + (−0.284 − 0.958i)9-s + 0.894i·10-s + (−0.288 − 0.408i)12-s + (−0.691 − 0.951i)13-s + (0.359 − 0.116i)14-s + (0.375 + 1.20i)15-s + (0.202 + 0.146i)16-s + (−1.17 − 0.855i)17-s + (0.561 + 0.430i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.622 + 0.782i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.622 + 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.463463 - 0.223647i\)
\(L(\frac12)\) \(\approx\) \(0.463463 - 0.223647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.03 - 1.38i)T \)
11 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.66 + 2.28i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.34 + 0.437i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.49 + 3.43i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.85 + 3.52i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.03 + 1.31i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (0.618 - 1.90i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.61 + 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.47 + 7.60i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.85 - 5.70i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 + (2.68 - 0.874i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.32 + 4.57i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (10.7 + 3.49i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.81 - 8.00i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (-1.66 + 2.28i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.34 + 0.437i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.49 + 3.43i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-12.9 - 9.40i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-1.61 + 1.17i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.19531343495457804157868428661, −9.955188272773366361102250709602, −9.431938346551469587518735462666, −8.847856517734985529585962690566, −7.58392431624018028128944135255, −6.49269423836943464697055956243, −5.35786639751846779875288672988, −4.51846021838089340681344488323, −3.09320112531718476959431527089, −0.44736532433448732277083633163, 1.71376286035673470351072787189, 2.68006223524200454241509391923, 4.85565252064632847934521381546, 6.18336365145869121124574180868, 6.48400505893399663796340634157, 7.74397280150646444696857822077, 9.065721450383967611744839841001, 9.875624556561101251449898475376, 10.64390727563127942649120848495, 11.36182678766866339292560608552

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