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Properties

Label 2-363-33.26-c2-0-33
Degree $2$
Conductor $363$
Sign $0.609 - 0.792i$
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  + (2.34 − 1.86i)3-s + (1.23 + 3.80i)4-s + (5.84 + 8.04i)5-s + (2.04 − 8.76i)9-s + (10 + 6.63i)12-s + (28.7 + 8.00i)15-s + (−12.9 + 9.40i)16-s + (−23.3 + 32.1i)20-s − 29.8i·23-s + (−22.8 + 70.3i)25-s + (−11.5 − 24.4i)27-s + (−29.9 − 21.7i)31-s + (35.8 − 3.06i)36-s + (7.72 + 23.7i)37-s + (82.5 − 34.8i)45-s + ⋯
L(s)  = 1  + (0.783 − 0.621i)3-s + (0.309 + 0.951i)4-s + (1.16 + 1.60i)5-s + (0.226 − 0.973i)9-s + (0.833 + 0.552i)12-s + (1.91 + 0.533i)15-s + (−0.809 + 0.587i)16-s + (−1.16 + 1.60i)20-s − 1.29i·23-s + (−0.914 + 2.81i)25-s + (−0.427 − 0.903i)27-s + (−0.965 − 0.701i)31-s + (0.996 − 0.0851i)36-s + (0.208 + 0.642i)37-s + (1.83 − 0.773i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.609 - 0.792i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.609 - 0.792i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.47241 + 1.21815i\)
\(L(\frac12)\) \(\approx\) \(2.47241 + 1.21815i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.34 + 1.86i)T \)
11 \( 1 \)
good2 \( 1 + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (-5.84 - 8.04i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (52.2 + 160. i)T^{2} \)
17 \( 1 + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-292. - 212. i)T^{2} \)
23 \( 1 + 29.8iT - 529T^{2} \)
29 \( 1 + (680. - 494. i)T^{2} \)
31 \( 1 + (29.9 + 21.7i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-7.72 - 23.7i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 + (-75.7 - 24.5i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-46.7 + 64.3i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (-47.3 + 15.3i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 35T + 4.48e3T^{2} \)
71 \( 1 + (29.2 + 40.2i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 149. iT - 7.92e3T^{2} \)
97 \( 1 + (76.8 + 55.8i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−11.31951419668425444316461715947, −10.43132437080144777362433318712, −9.462831928097351315550554175128, −8.465382909489005979358331438314, −7.34997559222228888553653032891, −6.82749136996301621792896343509, −5.94983882379685609032748647286, −3.80467210946420584630126957450, −2.76295687198298987409464097500, −2.11901223082422016696535796897, 1.27222538950813838168106715699, 2.30984882892034941942058075176, 4.21165340113189725491850466667, 5.29653504698049851881658378558, 5.76859396062668051806916100268, 7.39563826510653226332942860985, 8.799547743580634302255940626217, 9.192872787659393363025775429950, 9.974703799902601282827680593311, 10.71121363362206870730039668767

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