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Properties

Label 2-363-3.2-c2-0-4
Degree $2$
Conductor $363$
Sign $-0.701 + 0.713i$
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  + 2.87i·2-s + (2.10 − 2.13i)3-s − 4.28·4-s + 4.94i·5-s + (6.15 + 6.05i)6-s − 11.1·7-s − 0.807i·8-s + (−0.153 − 8.99i)9-s − 14.2·10-s + (−9.00 + 9.15i)12-s − 16.8·13-s − 32.1i·14-s + (10.5 + 10.4i)15-s − 14.7·16-s + 0.0625i·17-s + (25.8 − 0.441i)18-s + ⋯
L(s)  = 1  + 1.43i·2-s + (0.701 − 0.713i)3-s − 1.07·4-s + 0.989i·5-s + (1.02 + 1.00i)6-s − 1.59·7-s − 0.100i·8-s + (−0.0170 − 0.999i)9-s − 1.42·10-s + (−0.750 + 0.763i)12-s − 1.29·13-s − 2.29i·14-s + (0.705 + 0.693i)15-s − 0.924·16-s + 0.00368i·17-s + (1.43 − 0.0245i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.713i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.701 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.701 + 0.713i$
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: \((2,\ 363,\ (\ :1),\ -0.701 + 0.713i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.270877 - 0.646154i\)
\(L(\frac12)\) \(\approx\) \(0.270877 - 0.646154i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.10 + 2.13i)T \)
11 \( 1 \)
good2 \( 1 - 2.87iT - 4T^{2} \)
5 \( 1 - 4.94iT - 25T^{2} \)
7 \( 1 + 11.1T + 49T^{2} \)
13 \( 1 + 16.8T + 169T^{2} \)
17 \( 1 - 0.0625iT - 289T^{2} \)
19 \( 1 - 5.38T + 361T^{2} \)
23 \( 1 - 8.69iT - 529T^{2} \)
29 \( 1 - 49.9iT - 841T^{2} \)
31 \( 1 + 24.5T + 961T^{2} \)
37 \( 1 + 39.2T + 1.36e3T^{2} \)
41 \( 1 + 43.9iT - 1.68e3T^{2} \)
43 \( 1 - 0.201T + 1.84e3T^{2} \)
47 \( 1 - 14.9iT - 2.20e3T^{2} \)
53 \( 1 - 97.9iT - 2.80e3T^{2} \)
59 \( 1 - 14.4iT - 3.48e3T^{2} \)
61 \( 1 - 29.3T + 3.72e3T^{2} \)
67 \( 1 - 82.8T + 4.48e3T^{2} \)
71 \( 1 + 26.7iT - 5.04e3T^{2} \)
73 \( 1 - 13.9T + 5.32e3T^{2} \)
79 \( 1 - 50.1T + 6.24e3T^{2} \)
83 \( 1 - 60.2iT - 6.88e3T^{2} \)
89 \( 1 + 40.3iT - 7.92e3T^{2} \)
97 \( 1 + 30.8T + 9.40e3T^{2} \)
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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

−12.15698606268576749249033231458, −10.64769770458864839125123739577, −9.534242541410117200387573264184, −8.885325671679967737812719957342, −7.52950875094188744654289178075, −7.04278653935496513221607674740, −6.52519259193743023563435106237, −5.41304635506195144533980762472, −3.52394252062780173723223396877, −2.54984186783009059999091093090, 0.26579539377842926067463663120, 2.22496887355170563714154560775, 3.21364218912886372569226107550, 4.16073787700388900582374790560, 5.20188493347255146012569149606, 6.86852351018935794829129452282, 8.296551794939434476857314402994, 9.421903852425285083341368775596, 9.649605561251684230415045957284, 10.37656981568974118951385394316

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