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Properties

Label 2-363-33.5-c2-0-47
Degree $2$
Conductor $363$
Sign $-0.767 + 0.640i$
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  + (−2.06 + 0.670i)2-s + (1.56 − 2.56i)3-s + (0.570 − 0.414i)4-s + (−5.06 − 1.64i)5-s + (−1.50 + 6.32i)6-s + (10.0 − 7.32i)7-s + (4.20 − 5.78i)8-s + (−4.11 − 8.00i)9-s + 11.5·10-s + (−0.169 − 2.10i)12-s + (1.19 + 3.66i)13-s + (−15.8 + 21.8i)14-s + (−12.1 + 10.4i)15-s + (−5.66 + 17.4i)16-s + (−9.77 − 3.17i)17-s + (13.8 + 13.7i)18-s + ⋯
L(s)  = 1  + (−1.03 + 0.335i)2-s + (0.521 − 0.853i)3-s + (0.142 − 0.103i)4-s + (−1.01 − 0.329i)5-s + (−0.251 + 1.05i)6-s + (1.44 − 1.04i)7-s + (0.525 − 0.722i)8-s + (−0.456 − 0.889i)9-s + 1.15·10-s + (−0.0141 − 0.175i)12-s + (0.0915 + 0.281i)13-s + (−1.13 + 1.56i)14-s + (−0.809 + 0.693i)15-s + (−0.353 + 1.08i)16-s + (−0.574 − 0.186i)17-s + (0.769 + 0.764i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.767 + 0.640i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.767 + 0.640i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.252084 - 0.695504i\)
\(L(\frac12)\) \(\approx\) \(0.252084 - 0.695504i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.56 + 2.56i)T \)
11 \( 1 \)
good2 \( 1 + (2.06 - 0.670i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (5.06 + 1.64i)T + (20.2 + 14.6i)T^{2} \)
7 \( 1 + (-10.0 + 7.32i)T + (15.1 - 46.6i)T^{2} \)
13 \( 1 + (-1.19 - 3.66i)T + (-136. + 99.3i)T^{2} \)
17 \( 1 + (9.77 + 3.17i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (-17.8 - 12.9i)T + (111. + 343. i)T^{2} \)
23 \( 1 + 22.3iT - 529T^{2} \)
29 \( 1 + (6.37 + 8.77i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (5.84 + 17.9i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-4.62 + 3.35i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-15.2 + 21.0i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 + 62.1T + 1.84e3T^{2} \)
47 \( 1 + (33.5 - 46.1i)T + (-682. - 2.10e3i)T^{2} \)
53 \( 1 + (53.0 - 17.2i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (39.4 + 54.2i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-6.28 + 19.3i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 88.3T + 4.48e3T^{2} \)
71 \( 1 + (-101. - 32.9i)T + (4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-15.9 + 11.5i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (11.0 + 33.9i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-77.3 - 25.1i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 120. iT - 7.92e3T^{2} \)
97 \( 1 + (26.3 + 81.1i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−10.90111499237314113307204477381, −9.640181547842407031790811309449, −8.608798912255612589899176699084, −7.86562662041215990770972608127, −7.66796352610429558453535016274, −6.61560844581589475610135713008, −4.65182064967989832904936033721, −3.75826159241936761063565123339, −1.61065030068187590039677863304, −0.47195122779044073423859645925, 1.80327429391994776843754865107, 3.18274105436212804379290416114, 4.66096105803835005921184138400, 5.35072744027702669652332062542, 7.50447089151224523629246467028, 8.150420927619642697645772740299, 8.803355017889542620474541417981, 9.563649849166216754951046344205, 10.69642575590172145541460520745, 11.37104042918409888331951622379

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