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Properties

Label 2-363-11.3-c1-0-9
Degree $2$
Conductor $363$
Sign $0.751 + 0.659i$
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  + (−2.11 + 1.53i)2-s + (−0.309 − 0.951i)3-s + (1.5 − 4.61i)4-s + (0.5 + 0.363i)5-s + (2.11 + 1.53i)6-s + (0.309 − 0.951i)7-s + (2.30 + 7.10i)8-s + (−0.809 + 0.587i)9-s − 1.61·10-s − 4.85·12-s + (0.190 − 0.138i)13-s + (0.809 + 2.48i)14-s + (0.190 − 0.587i)15-s + (−7.97 − 5.79i)16-s + (−0.927 − 0.673i)17-s + (0.809 − 2.48i)18-s + ⋯
L(s)  = 1  + (−1.49 + 1.08i)2-s + (−0.178 − 0.549i)3-s + (0.750 − 2.30i)4-s + (0.223 + 0.162i)5-s + (0.864 + 0.628i)6-s + (0.116 − 0.359i)7-s + (0.816 + 2.51i)8-s + (−0.269 + 0.195i)9-s − 0.511·10-s − 1.40·12-s + (0.0529 − 0.0384i)13-s + (0.216 + 0.665i)14-s + (0.0493 − 0.151i)15-s + (−1.99 − 1.44i)16-s + (−0.224 − 0.163i)17-s + (0.190 − 0.586i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489157 - 0.184075i\)
\(L(\frac12)\) \(\approx\) \(0.489157 - 0.184075i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (2.11 - 1.53i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.5 - 0.363i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.190 + 0.138i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.927 + 0.673i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.236T + 23T^{2} \)
29 \( 1 + (-1.85 + 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.92 + 3.57i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.92 - 5.93i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.0729 + 0.224i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.70T + 43T^{2} \)
47 \( 1 + (3.11 + 9.59i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.309 + 0.224i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.28 + 7.02i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (9.35 + 6.79i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 1.85T + 67T^{2} \)
71 \( 1 + (8.35 + 6.06i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.76 + 5.42i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.89 + 6.46i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.19 - 0.865i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.23T + 89T^{2} \)
97 \( 1 + (6.35 - 4.61i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.96261400251855302843839548763, −10.20024449882396976510623672915, −9.297226099426997506362517302827, −8.364320332696430081101218688701, −7.63068183853984660780592469763, −6.67875773645983708311433837772, −6.12797290465361444543822878569, −4.78225054740279230987821264256, −2.26750602750295708225563223956, −0.61893711627629929726786056548, 1.51156156693103737334209707350, 2.90379195700167494169822798263, 4.12845160631002208742763406072, 5.73836767489756991819298644282, 7.18073510245233359133889444389, 8.314039392246169966491289403067, 8.952095025784348930146286375358, 9.772333479153175490631149122071, 10.54404839818605914599995620682, 11.17638069045067913678476796520

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