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Properties

Label 2-363-33.26-c2-0-47
Degree $2$
Conductor $363$
Sign $0.478 + 0.878i$
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  + (0.974 − 1.34i)2-s + (2.52 − 1.61i)3-s + (0.386 + 1.18i)4-s + (0.410 + 0.565i)5-s + (0.300 − 4.96i)6-s + (−0.806 − 2.48i)7-s + (8.28 + 2.69i)8-s + (3.79 − 8.16i)9-s + 1.15·10-s + (2.89 + 2.38i)12-s + (13.8 + 10.0i)13-s + (−4.11 − 1.33i)14-s + (1.95 + 0.766i)15-s + (7.63 − 5.54i)16-s + (−9.47 − 13.0i)17-s + (−7.25 − 13.0i)18-s + ⋯
L(s)  = 1  + (0.487 − 0.670i)2-s + (0.843 − 0.537i)3-s + (0.0966 + 0.297i)4-s + (0.0821 + 0.113i)5-s + (0.0500 − 0.827i)6-s + (−0.115 − 0.354i)7-s + (1.03 + 0.336i)8-s + (0.421 − 0.906i)9-s + 0.115·10-s + (0.241 + 0.198i)12-s + (1.06 + 0.773i)13-s + (−0.293 − 0.0954i)14-s + (0.130 + 0.0511i)15-s + (0.477 − 0.346i)16-s + (−0.557 − 0.766i)17-s + (−0.402 − 0.724i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.478 + 0.878i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.478 + 0.878i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.76273 - 1.64179i\)
\(L(\frac12)\) \(\approx\) \(2.76273 - 1.64179i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.52 + 1.61i)T \)
11 \( 1 \)
good2 \( 1 + (-0.974 + 1.34i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (-0.410 - 0.565i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (0.806 + 2.48i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (-13.8 - 10.0i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (9.47 + 13.0i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-4.92 + 15.1i)T + (-292. - 212. i)T^{2} \)
23 \( 1 - 23.1iT - 529T^{2} \)
29 \( 1 + (5.10 - 1.65i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-3.28 - 2.38i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-19.6 - 60.4i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (64.1 + 20.8i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 22.6T + 1.84e3T^{2} \)
47 \( 1 + (70.2 + 22.8i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (25.1 - 34.6i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (27.3 - 8.88i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (37.4 - 27.2i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 77.2T + 4.48e3T^{2} \)
71 \( 1 + (-24.2 - 33.3i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-17.5 - 53.9i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (41.1 + 29.9i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-34.8 - 47.9i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 38.1iT - 7.92e3T^{2} \)
97 \( 1 + (13.1 + 9.57i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−11.40203228099089204024064323105, −10.24283988843027861216571287986, −9.130571523962317133601196077437, −8.284602205593244726060616039488, −7.23887787567053493959268907108, −6.51566144842913839394473253901, −4.68235057279938144653518088194, −3.63862297691210574709227960801, −2.71802474960887502227126960641, −1.45202324460079378358063871590, 1.71405545678666072810966990178, 3.34712182645287563346407492822, 4.44658626279380094284275867733, 5.54294301848925304996854449323, 6.39020763178355073296819758158, 7.68107025328415828317330565382, 8.479749522176149228143870543291, 9.456177617888327877222284194533, 10.46743611654009824264415843924, 11.02815512997258869741137058151

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