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Properties

Label 2-363-33.20-c2-0-21
Degree $2$
Conductor $363$
Sign $0.584 - 0.811i$
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  + (−2.51 − 0.817i)2-s + (2.47 + 1.70i)3-s + (2.42 + 1.76i)4-s + (2.68 − 0.874i)5-s + (−4.82 − 6.29i)6-s + (−6.05 − 4.39i)7-s + (1.55 + 2.14i)8-s + (3.21 + 8.40i)9-s − 7.48·10-s + (2.99 + 8.48i)12-s + (−6.93 + 21.3i)13-s + (11.6 + 16.0i)14-s + (8.13 + 2.41i)15-s + (−5.87 − 18.0i)16-s + (20.1 − 6.54i)17-s + (−1.22 − 23.7i)18-s + ⋯
L(s)  = 1  + (−1.25 − 0.408i)2-s + (0.823 + 0.566i)3-s + (0.606 + 0.440i)4-s + (0.537 − 0.174i)5-s + (−0.804 − 1.04i)6-s + (−0.864 − 0.628i)7-s + (0.194 + 0.267i)8-s + (0.357 + 0.933i)9-s − 0.748·10-s + (0.249 + 0.707i)12-s + (−0.533 + 1.64i)13-s + (0.831 + 1.14i)14-s + (0.542 + 0.160i)15-s + (−0.366 − 1.12i)16-s + (1.18 − 0.384i)17-s + (−0.0679 − 1.32i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.584 - 0.811i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.584 - 0.811i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.911681 + 0.466646i\)
\(L(\frac12)\) \(\approx\) \(0.911681 + 0.466646i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.47 - 1.70i)T \)
11 \( 1 \)
good2 \( 1 + (2.51 + 0.817i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (-2.68 + 0.874i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (6.05 + 4.39i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (6.93 - 21.3i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-20.1 + 6.54i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-12.1 + 8.79i)T + (111. - 343. i)T^{2} \)
23 \( 1 - 31.1iT - 529T^{2} \)
29 \( 1 + (12.4 - 17.1i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-9.27 + 28.5i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-8.09 - 5.87i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-24.8 - 34.2i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 14.9T + 1.84e3T^{2} \)
47 \( 1 + (-21.6 - 29.7i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (-40.3 - 13.1i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (19.9 - 27.4i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-30.0 - 92.5i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 42T + 4.48e3T^{2} \)
71 \( 1 + (-61.8 + 20.1i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-60.5 - 43.9i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (6.93 - 21.3i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-20.1 + 6.54i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 62.2iT - 7.92e3T^{2} \)
97 \( 1 + (-22.8 + 70.3i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.01609667044787534797212232093, −9.808307169363713632770223559899, −9.640606567980661080446665298295, −9.092485743898913232393787014156, −7.74306971522598365150202976519, −7.13206335638599196367526947969, −5.40706428938992422466168682365, −4.06282511340828046159954464501, −2.73288899535779264820683473404, −1.41120219775810175717243773528, 0.68499471031024490085113598247, 2.35967932085139200270698675044, 3.50346122830794165494886913507, 5.68598326466397916052387589902, 6.55277442222439434506211945038, 7.66801534439593766554838837606, 8.193035345602369466663451370955, 9.193391600478205741775183194284, 9.926926285202188155162954840457, 10.40622468100901689486956813490

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