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Properties

Label 2-363-3.2-c2-0-12
Degree $2$
Conductor $363$
Sign $-0.333 - 0.942i$
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  + 2.64i·2-s + (−1 − 2.82i)3-s − 3.00·4-s − 2.82i·5-s + (7.48 − 2.64i)6-s − 7.48·7-s + 2.64i·8-s + (−7.00 + 5.65i)9-s + 7.48·10-s + (3.00 + 8.48i)12-s + 22.4·13-s − 19.7i·14-s + (−8.00 + 2.82i)15-s − 18.9·16-s + 21.1i·17-s + (−14.9 − 18.5i)18-s + ⋯
L(s)  = 1  + 1.32i·2-s + (−0.333 − 0.942i)3-s − 0.750·4-s − 0.565i·5-s + (1.24 − 0.440i)6-s − 1.06·7-s + 0.330i·8-s + (−0.777 + 0.628i)9-s + 0.748·10-s + (0.250 + 0.707i)12-s + 1.72·13-s − 1.41i·14-s + (−0.533 + 0.188i)15-s − 1.18·16-s + 1.24i·17-s + (−0.831 − 1.02i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.333 - 0.942i$
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: \((2,\ 363,\ (\ :1),\ -0.333 - 0.942i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.698451 + 0.987760i\)
\(L(\frac12)\) \(\approx\) \(0.698451 + 0.987760i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1 + 2.82i)T \)
11 \( 1 \)
good2 \( 1 - 2.64iT - 4T^{2} \)
5 \( 1 + 2.82iT - 25T^{2} \)
7 \( 1 + 7.48T + 49T^{2} \)
13 \( 1 - 22.4T + 169T^{2} \)
17 \( 1 - 21.1iT - 289T^{2} \)
19 \( 1 - 14.9T + 361T^{2} \)
23 \( 1 - 31.1iT - 529T^{2} \)
29 \( 1 - 21.1iT - 841T^{2} \)
31 \( 1 - 30T + 961T^{2} \)
37 \( 1 + 10T + 1.36e3T^{2} \)
41 \( 1 - 42.3iT - 1.68e3T^{2} \)
43 \( 1 + 14.9T + 1.84e3T^{2} \)
47 \( 1 + 36.7iT - 2.20e3T^{2} \)
53 \( 1 - 42.4iT - 2.80e3T^{2} \)
59 \( 1 + 33.9iT - 3.48e3T^{2} \)
61 \( 1 + 97.2T + 3.72e3T^{2} \)
67 \( 1 + 42T + 4.48e3T^{2} \)
71 \( 1 + 65.0iT - 5.04e3T^{2} \)
73 \( 1 - 74.8T + 5.32e3T^{2} \)
79 \( 1 - 22.4T + 6.24e3T^{2} \)
83 \( 1 - 21.1iT - 6.88e3T^{2} \)
89 \( 1 - 62.2iT - 7.92e3T^{2} \)
97 \( 1 - 74T + 9.40e3T^{2} \)
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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

−11.62490655802646324267738186163, −10.65646997521070812744295205351, −9.168034924062562742060587603924, −8.415983371986487449779387551532, −7.61853625544454035444026268708, −6.47338713524486528421817897006, −6.12873561599567231470033137604, −5.11849252463177757637651871015, −3.37266882361904690293500604173, −1.33801316401957136952610428961, 0.63650292951615646684212584155, 2.83730532132860287924391173826, 3.40926173386547506982743245182, 4.51870337630517113599828259052, 6.05723558002280440820479234604, 6.83737207397697500399098337812, 8.690600173452106248225687545348, 9.511584420427842835655536365645, 10.23901363043048731644885255830, 10.88733166681759000455739103118

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