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Properties

Label 2-363-3.2-c2-0-41
Degree $2$
Conductor $363$
Sign $0.666 + 0.745i$
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  + i·2-s + (−2 − 2.23i)3-s + 3·4-s − 6.61i·5-s + (2.23 − 2i)6-s + 8.85·7-s + 7i·8-s + (−1.00 + 8.94i)9-s + 6.61·10-s + (−6 − 6.70i)12-s + 13.2·13-s + 8.85i·14-s + (−14.7 + 13.2i)15-s + 5·16-s − 6i·17-s + (−8.94 − 1.00i)18-s + ⋯
L(s)  = 1  + 0.5i·2-s + (−0.666 − 0.745i)3-s + 0.750·4-s − 1.32i·5-s + (0.372 − 0.333i)6-s + 1.26·7-s + 0.875i·8-s + (−0.111 + 0.993i)9-s + 0.661·10-s + (−0.5 − 0.559i)12-s + 1.01·13-s + 0.632i·14-s + (−0.986 + 0.882i)15-s + 0.312·16-s − 0.352i·17-s + (−0.496 − 0.0555i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.666 + 0.745i$
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: \((2,\ 363,\ (\ :1),\ 0.666 + 0.745i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.74220 - 0.779137i\)
\(L(\frac12)\) \(\approx\) \(1.74220 - 0.779137i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2 + 2.23i)T \)
11 \( 1 \)
good2 \( 1 - iT - 4T^{2} \)
5 \( 1 + 6.61iT - 25T^{2} \)
7 \( 1 - 8.85T + 49T^{2} \)
13 \( 1 - 13.2T + 169T^{2} \)
17 \( 1 + 6iT - 289T^{2} \)
19 \( 1 + 9.12T + 361T^{2} \)
23 \( 1 + 17.5iT - 529T^{2} \)
29 \( 1 + 26.3iT - 841T^{2} \)
31 \( 1 + 6.20T + 961T^{2} \)
37 \( 1 + 19.5T + 1.36e3T^{2} \)
41 \( 1 + 5.59iT - 1.68e3T^{2} \)
43 \( 1 - 26.2T + 1.84e3T^{2} \)
47 \( 1 + 17.7iT - 2.20e3T^{2} \)
53 \( 1 - 22.2iT - 2.80e3T^{2} \)
59 \( 1 - 21.9iT - 3.48e3T^{2} \)
61 \( 1 - 93.3T + 3.72e3T^{2} \)
67 \( 1 + 76.7T + 4.48e3T^{2} \)
71 \( 1 + 65.8iT - 5.04e3T^{2} \)
73 \( 1 + 15.0T + 5.32e3T^{2} \)
79 \( 1 + 118.T + 6.24e3T^{2} \)
83 \( 1 - 111. iT - 6.88e3T^{2} \)
89 \( 1 + 97.6iT - 7.92e3T^{2} \)
97 \( 1 + 121.T + 9.40e3T^{2} \)
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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

−11.31424669190434238303083668423, −10.51271480701794973394718789695, −8.705298773501913905108487269484, −8.218183275903349848593917168353, −7.33194239174761327150974225427, −6.18726138607241031963096552867, −5.38272138944625420677288373937, −4.50317529409043112123234048329, −2.09838360664788869631182814085, −1.05641556111361847549559825030, 1.58129250509582049655980319149, 3.10551364849772537146640581537, 4.07817993899722393754091457722, 5.55704372508261966117570171207, 6.48532983372979218414150769508, 7.35895270516037598738153910744, 8.643583646926852779030902030302, 10.02763479217110465797872957305, 10.73064353525715774272971615437, 11.18401756493805968097814748554

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