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Properties

Label 2-363-3.2-c2-0-33
Degree $2$
Conductor $363$
Sign $0.895 + 0.445i$
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  + 2.52i·2-s + (−2.68 − 1.33i)3-s − 2.37·4-s − 0.792i·5-s + (3.37 − 6.78i)6-s − 6.74·7-s + 4.10i·8-s + (5.43 + 7.17i)9-s + 2·10-s + (6.37 + 3.16i)12-s − 9.48·13-s − 17.0i·14-s + (−1.05 + 2.12i)15-s − 19.8·16-s − 29.2i·17-s + (−18.1 + 13.7i)18-s + ⋯
L(s)  = 1  + 1.26i·2-s + (−0.895 − 0.445i)3-s − 0.593·4-s − 0.158i·5-s + (0.562 − 1.13i)6-s − 0.963·7-s + 0.513i·8-s + (0.603 + 0.797i)9-s + 0.200·10-s + (0.531 + 0.264i)12-s − 0.729·13-s − 1.21i·14-s + (−0.0705 + 0.141i)15-s − 1.24·16-s − 1.72i·17-s + (−1.00 + 0.761i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.895 + 0.445i$
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: \((2,\ 363,\ (\ :1),\ 0.895 + 0.445i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.698573 - 0.164123i\)
\(L(\frac12)\) \(\approx\) \(0.698573 - 0.164123i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.68 + 1.33i)T \)
11 \( 1 \)
good2 \( 1 - 2.52iT - 4T^{2} \)
5 \( 1 + 0.792iT - 25T^{2} \)
7 \( 1 + 6.74T + 49T^{2} \)
13 \( 1 + 9.48T + 169T^{2} \)
17 \( 1 + 29.2iT - 289T^{2} \)
19 \( 1 - 26.2T + 361T^{2} \)
23 \( 1 + 26.9iT - 529T^{2} \)
29 \( 1 + 25.9iT - 841T^{2} \)
31 \( 1 + 2.86T + 961T^{2} \)
37 \( 1 + 2.39T + 1.36e3T^{2} \)
41 \( 1 + 17.6iT - 1.68e3T^{2} \)
43 \( 1 + 12.5T + 1.84e3T^{2} \)
47 \( 1 - 41.6iT - 2.20e3T^{2} \)
53 \( 1 + 89.5iT - 2.80e3T^{2} \)
59 \( 1 + 14.7iT - 3.48e3T^{2} \)
61 \( 1 - 63.4T + 3.72e3T^{2} \)
67 \( 1 + 63.3T + 4.48e3T^{2} \)
71 \( 1 - 4.55iT - 5.04e3T^{2} \)
73 \( 1 - 53.7T + 5.32e3T^{2} \)
79 \( 1 + 55.6T + 6.24e3T^{2} \)
83 \( 1 + 65.3iT - 6.88e3T^{2} \)
89 \( 1 - 14.1iT - 7.92e3T^{2} \)
97 \( 1 - 149.T + 9.40e3T^{2} \)
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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

−11.35447600092993361382210007156, −10.07265558416030311213574805228, −9.224306440404642023406822311245, −7.898725675774542855132189876123, −7.05943048157780258827524825755, −6.55866897057035480557951979891, −5.43876514375927257514435125039, −4.77089283749589369907728052037, −2.68014588804427720236458805973, −0.39065716315635361243385313878, 1.30165593428722335016829528314, 3.08011797297600168016539549896, 3.88620190720541997643853030273, 5.24573634063919200640776344882, 6.38752676768062143798355637348, 7.28742796022470551230842081172, 9.089064479467122617320847117465, 9.868701546121122765571373541939, 10.42219297233024035897860205433, 11.22809234999847040762157116965

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