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Properties

Label 2-363-11.9-c1-0-8
Degree $2$
Conductor $363$
Sign $0.944 + 0.329i$
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  + (0.190 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (1.30 + 0.951i)4-s + (−0.809 − 2.48i)5-s + (0.190 + 0.587i)6-s + (2.42 + 1.76i)7-s + (1.80 − 1.31i)8-s + (0.309 − 0.951i)9-s − 1.61·10-s − 1.61·12-s + (−0.545 + 1.67i)13-s + (1.5 − 1.08i)14-s + (2.11 + 1.53i)15-s + (0.572 + 1.76i)16-s + (−0.5 − 1.53i)17-s + (−0.5 − 0.363i)18-s + ⋯
L(s)  = 1  + (0.135 − 0.415i)2-s + (−0.467 + 0.339i)3-s + (0.654 + 0.475i)4-s + (−0.361 − 1.11i)5-s + (0.0779 + 0.239i)6-s + (0.917 + 0.666i)7-s + (0.639 − 0.464i)8-s + (0.103 − 0.317i)9-s − 0.511·10-s − 0.467·12-s + (−0.151 + 0.465i)13-s + (0.400 − 0.291i)14-s + (0.546 + 0.397i)15-s + (0.143 + 0.440i)16-s + (−0.121 − 0.373i)17-s + (−0.117 − 0.0856i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.944 + 0.329i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53544 - 0.260524i\)
\(L(\frac12)\) \(\approx\) \(1.53544 - 0.260524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-0.190 + 0.587i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.809 + 2.48i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.42 - 1.76i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.545 - 1.67i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.73 + 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (-3.61 - 2.62i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.881 + 2.71i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.190 + 0.138i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.66 - 7.02i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.23T + 43T^{2} \)
47 \( 1 + (1.30 - 0.951i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.97 - 9.14i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.35 + 6.06i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.42 + 7.46i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 9.56T + 67T^{2} \)
71 \( 1 + (1.71 + 5.29i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.61 + 1.90i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.92 - 9.00i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.218 + 0.673i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 0.527T + 89T^{2} \)
97 \( 1 + (4.33 - 13.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.68234759032165547880557836736, −10.80890217384698239170143399636, −9.513340246454958810914141499282, −8.629486171163322095248239057448, −7.72623546436009186779480678521, −6.59401636727049831138710924878, −5.07337895770830217275557492218, −4.60851830779525113903064995186, −3.05223612619032627501002467731, −1.43607979429484885392370960091, 1.51756182927486248169965921219, 3.14997464555285877804653587573, 4.77994006720615815586308854130, 5.79045300097549793679347253391, 6.88360530153483198766165017126, 7.37859679750734524258532826405, 8.235321082445680556134545230173, 10.18552226632240301940468707306, 10.58556190447344945042600258684, 11.42625515564436929233262615982

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