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Properties

Label 2-363-3.2-c2-0-40
Degree $2$
Conductor $363$
Sign $0.772 + 0.635i$
Analytic cond. $9.89103$
Root an. cond. $3.14500$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.65i·2-s + (2.31 + 1.90i)3-s + 1.25·4-s − 0.698i·5-s + (3.16 − 3.84i)6-s + 2.60·7-s − 8.70i·8-s + (1.72 + 8.83i)9-s − 1.15·10-s + (2.89 + 2.38i)12-s + 17.1·13-s − 4.32i·14-s + (1.33 − 1.61i)15-s − 9.43·16-s − 16.1i·17-s + (14.6 − 2.86i)18-s + ⋯
L(s)  = 1  − 0.829i·2-s + (0.772 + 0.635i)3-s + 0.312·4-s − 0.139i·5-s + (0.526 − 0.640i)6-s + 0.372·7-s − 1.08i·8-s + (0.192 + 0.981i)9-s − 0.115·10-s + (0.241 + 0.198i)12-s + 1.31·13-s − 0.309i·14-s + (0.0887 − 0.107i)15-s − 0.589·16-s − 0.947i·17-s + (0.813 − 0.159i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.772 + 0.635i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.772 + 0.635i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.54008 - 0.910958i\)
\(L(\frac12)\) \(\approx\) \(2.54008 - 0.910958i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.31 - 1.90i)T \)
11 \( 1 \)
good2 \( 1 + 1.65iT - 4T^{2} \)
5 \( 1 + 0.698iT - 25T^{2} \)
7 \( 1 - 2.60T + 49T^{2} \)
13 \( 1 - 17.1T + 169T^{2} \)
17 \( 1 + 16.1iT - 289T^{2} \)
19 \( 1 + 15.9T + 361T^{2} \)
23 \( 1 - 23.1iT - 529T^{2} \)
29 \( 1 + 5.36iT - 841T^{2} \)
31 \( 1 + 4.06T + 961T^{2} \)
37 \( 1 - 63.5T + 1.36e3T^{2} \)
41 \( 1 - 67.4iT - 1.68e3T^{2} \)
43 \( 1 + 22.6T + 1.84e3T^{2} \)
47 \( 1 + 73.8iT - 2.20e3T^{2} \)
53 \( 1 + 42.8iT - 2.80e3T^{2} \)
59 \( 1 - 28.7iT - 3.48e3T^{2} \)
61 \( 1 + 46.3T + 3.72e3T^{2} \)
67 \( 1 + 77.2T + 4.48e3T^{2} \)
71 \( 1 + 41.2iT - 5.04e3T^{2} \)
73 \( 1 + 56.7T + 5.32e3T^{2} \)
79 \( 1 + 50.9T + 6.24e3T^{2} \)
83 \( 1 - 59.2iT - 6.88e3T^{2} \)
89 \( 1 + 38.1iT - 7.92e3T^{2} \)
97 \( 1 - 16.2T + 9.40e3T^{2} \)
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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

−11.14377308372269030877131195778, −10.27615932548633839112048341040, −9.422066227500574760083314951554, −8.541976768366996120470808973598, −7.54145267097791874022897484296, −6.29294258741258507248054315255, −4.80120482569513839545016727621, −3.71735900695389851761510610049, −2.74871458133415072275991495083, −1.43625806949560011018261247788, 1.55916667345525612497982650865, 2.84529142656888042313113446736, 4.29657862250094771026318062361, 5.99687608009064779004291041054, 6.51765018431180009396217571992, 7.57578324653428369209604284105, 8.383927458760950376149750532749, 8.907001552324668258316509143635, 10.54893521600205500877438773761, 11.21706155811415152002589692430

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