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Properties

Label 2-363-33.32-c1-0-15
Degree $2$
Conductor $363$
Sign $0.833 + 0.552i$
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  − 1.90·2-s + (1.61 + 0.618i)3-s + 1.61·4-s − 2.61i·5-s + (−3.07 − 1.17i)6-s + 0.726i·7-s + 0.726·8-s + (2.23 + 2.00i)9-s + 4.97i·10-s + (2.61 + 0.999i)12-s − 3.07i·13-s − 1.38i·14-s + (1.61 − 4.23i)15-s − 4.61·16-s + 2.62·17-s + (−4.25 − 3.80i)18-s + ⋯
L(s)  = 1  − 1.34·2-s + (0.934 + 0.356i)3-s + 0.809·4-s − 1.17i·5-s + (−1.25 − 0.479i)6-s + 0.274i·7-s + 0.256·8-s + (0.745 + 0.666i)9-s + 1.57i·10-s + (0.755 + 0.288i)12-s − 0.853i·13-s − 0.369i·14-s + (0.417 − 1.09i)15-s − 1.15·16-s + 0.637·17-s + (−1.00 − 0.896i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.833 + 0.552i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.833 + 0.552i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.903507 - 0.272432i\)
\(L(\frac12)\) \(\approx\) \(0.903507 - 0.272432i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.61 - 0.618i)T \)
11 \( 1 \)
good2 \( 1 + 1.90T + 2T^{2} \)
5 \( 1 + 2.61iT - 5T^{2} \)
7 \( 1 - 0.726iT - 7T^{2} \)
13 \( 1 + 3.07iT - 13T^{2} \)
17 \( 1 - 2.62T + 17T^{2} \)
19 \( 1 + 4.25iT - 19T^{2} \)
23 \( 1 + 1.76iT - 23T^{2} \)
29 \( 1 - 3.80T + 29T^{2} \)
31 \( 1 + 0.854T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 - 3.07T + 41T^{2} \)
43 \( 1 + 1.62iT - 43T^{2} \)
47 \( 1 + 7.32iT - 47T^{2} \)
53 \( 1 - 4.85iT - 53T^{2} \)
59 \( 1 + 2.61iT - 59T^{2} \)
61 \( 1 - 4.25iT - 61T^{2} \)
67 \( 1 + 8.32T + 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 - 15.2iT - 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 - 9.47iT - 89T^{2} \)
97 \( 1 + 6.61T + 97T^{2} \)
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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

−10.90479867707244494349162031246, −10.06310517462550578977403005718, −9.303649115417189356463858982645, −8.628602508929852529171193671038, −8.105059545989767369649608798803, −7.12926489809822006475952361768, −5.32768893637914041006426421035, −4.30629584074771619887681102996, −2.62358248263222631889263992220, −1.04940593346052354861744081710, 1.54196245060077991576343635049, 2.86555412724676102260942111615, 4.15301051889662570161886511287, 6.30864400612756689800342214033, 7.22809906259628934160691666067, 7.77769541116855828386172268862, 8.746007420101451851216286173564, 9.667804486586689186943383920058, 10.25803041749751075909725081478, 11.15824842437195575201376152728

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