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Properties

Label 2-363-3.2-c2-0-24
Degree $2$
Conductor $363$
Sign $0.0800 - 0.996i$
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  + 2.16i·2-s + (0.240 − 2.99i)3-s − 0.704·4-s + 5.33i·5-s + (6.48 + 0.521i)6-s + 12.4·7-s + 7.14i·8-s + (−8.88 − 1.43i)9-s − 11.5·10-s + (−0.169 + 2.10i)12-s − 3.85·13-s + 27.0i·14-s + (15.9 + 1.28i)15-s − 18.3·16-s − 10.2i·17-s + (3.11 − 19.2i)18-s + ⋯
L(s)  = 1  + 1.08i·2-s + (0.0800 − 0.996i)3-s − 0.176·4-s + 1.06i·5-s + (1.08 + 0.0868i)6-s + 1.78·7-s + 0.893i·8-s + (−0.987 − 0.159i)9-s − 1.15·10-s + (−0.0141 + 0.175i)12-s − 0.296·13-s + 1.93i·14-s + (1.06 + 0.0853i)15-s − 1.14·16-s − 0.604i·17-s + (0.173 − 1.07i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0800 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0800 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.0800 - 0.996i$
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: \((2,\ 363,\ (\ :1),\ 0.0800 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.59016 + 1.46751i\)
\(L(\frac12)\) \(\approx\) \(1.59016 + 1.46751i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.240 + 2.99i)T \)
11 \( 1 \)
good2 \( 1 - 2.16iT - 4T^{2} \)
5 \( 1 - 5.33iT - 25T^{2} \)
7 \( 1 - 12.4T + 49T^{2} \)
13 \( 1 + 3.85T + 169T^{2} \)
17 \( 1 + 10.2iT - 289T^{2} \)
19 \( 1 - 22.0T + 361T^{2} \)
23 \( 1 - 22.3iT - 529T^{2} \)
29 \( 1 - 10.8iT - 841T^{2} \)
31 \( 1 + 18.9T + 961T^{2} \)
37 \( 1 + 5.71T + 1.36e3T^{2} \)
41 \( 1 - 26.0iT - 1.68e3T^{2} \)
43 \( 1 - 62.1T + 1.84e3T^{2} \)
47 \( 1 - 57.0iT - 2.20e3T^{2} \)
53 \( 1 + 55.7iT - 2.80e3T^{2} \)
59 \( 1 + 67.0iT - 3.48e3T^{2} \)
61 \( 1 + 20.3T + 3.72e3T^{2} \)
67 \( 1 + 88.3T + 4.48e3T^{2} \)
71 \( 1 + 106. iT - 5.04e3T^{2} \)
73 \( 1 - 19.6T + 5.32e3T^{2} \)
79 \( 1 - 35.7T + 6.24e3T^{2} \)
83 \( 1 - 81.3iT - 6.88e3T^{2} \)
89 \( 1 + 120. iT - 7.92e3T^{2} \)
97 \( 1 + 85.3T + 9.40e3T^{2} \)
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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

−11.31227186567589627290059398815, −10.97532474286941165240430321521, −9.225919968195506721017820769628, −8.037350268867625071962169918825, −7.53241815512002512094306573905, −6.97473924876663426019099112332, −5.80344223123088404786462722297, −4.98378558938934765921806189885, −2.90023968614848375897142594721, −1.67130794751689602561877756480, 1.09317522984858839973945894734, 2.39678250071812401006176192362, 3.96976948263358506705787404585, 4.71562155025325005411653307976, 5.59933890534355250195130222344, 7.54029413998119709723099224560, 8.582710339371314570575941679371, 9.234144545320850307949480918460, 10.37128793192133702087839732685, 10.93290593035682220153181308877

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