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Properties

Label 2-363-33.2-c1-0-21
Degree $2$
Conductor $363$
Sign $0.913 + 0.407i$
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  + (1.40 − 1.01i)3-s + (1.61 + 1.17i)4-s + (1.35 − 1.86i)7-s + (0.927 − 2.85i)9-s + 3.46·12-s + (−6.00 − 1.95i)13-s + (1.23 + 3.80i)16-s + (5.06 + 6.97i)19-s − 4.00i·21-s + (−4.04 + 2.93i)25-s + (−1.60 − 4.94i)27-s + (4.39 − 1.42i)28-s + (0.535 − 1.64i)31-s + (4.85 − 3.52i)36-s + (−4.20 − 3.05i)37-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (0.809 + 0.587i)4-s + (0.513 − 0.706i)7-s + (0.309 − 0.951i)9-s + 1.00·12-s + (−1.66 − 0.541i)13-s + (0.309 + 0.951i)16-s + (1.16 + 1.60i)19-s − 0.873i·21-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (0.830 − 0.269i)28-s + (0.0961 − 0.295i)31-s + (0.809 − 0.587i)36-s + (−0.691 − 0.502i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.913 + 0.407i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.913 + 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99904 - 0.425773i\)
\(L(\frac12)\) \(\approx\) \(1.99904 - 0.425773i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.40 + 1.01i)T \)
11 \( 1 \)
good2 \( 1 + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.35 + 1.86i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (6.00 + 1.95i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.06 - 6.97i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.535 + 1.64i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.20 + 3.05i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.69iT - 43T^{2} \)
47 \( 1 + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.81 + 3.18i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.78 - 10.7i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.588 + 0.191i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (1.54 - 4.75i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.65375885379185018462038742228, −10.36744101314987015571734213507, −9.578017519853103536738903882683, −8.149963210480692380819781342760, −7.59865265777044347820282535177, −7.07261724729318081455987695531, −5.64305043106992866409939588559, −3.99237955885935371919297207637, −2.92744515612453546707806758427, −1.67841939446086799315874815955, 2.06147845641030485533190427657, 2.89054032166992973181515676859, 4.69242940288082217283610625496, 5.38862971553732293319462207176, 6.92645507012870163278708217779, 7.67346312112080205297653834055, 8.910308348801445538574670400747, 9.661476124236935386134363909502, 10.41164489516483152555549187365, 11.56294256723113033664155395787

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