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Properties

Label 2-363-33.14-c2-0-23
Degree $2$
Conductor $363$
Sign $-0.782 + 0.622i$
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  + (1.94 + 2.68i)2-s + (0.927 + 2.85i)3-s + (−2.16 + 6.65i)4-s + (−3.89 + 5.36i)5-s + (−5.84 + 8.04i)6-s + (2.47 − 7.60i)7-s + (−9.46 + 3.07i)8-s + (−7.28 + 5.29i)9-s − 22·10-s − 21.0·12-s + (3.23 − 2.35i)13-s + (25.2 − 8.19i)14-s + (−18.9 − 6.14i)15-s + (−4.04 − 2.93i)16-s + (−7.79 + 10.7i)17-s + (−28.3 − 9.22i)18-s + ⋯
L(s)  = 1  + (0.974 + 1.34i)2-s + (0.309 + 0.951i)3-s + (−0.540 + 1.66i)4-s + (−0.779 + 1.07i)5-s + (−0.974 + 1.34i)6-s + (0.353 − 1.08i)7-s + (−1.18 + 0.384i)8-s + (−0.809 + 0.587i)9-s − 2.20·10-s − 1.75·12-s + (0.248 − 0.180i)13-s + (1.80 − 0.585i)14-s + (−1.26 − 0.409i)15-s + (−0.252 − 0.183i)16-s + (−0.458 + 0.631i)17-s + (−1.57 − 0.512i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.782 + 0.622i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.782 + 0.622i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.808129 - 2.31287i\)
\(L(\frac12)\) \(\approx\) \(0.808129 - 2.31287i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.927 - 2.85i)T \)
11 \( 1 \)
good2 \( 1 + (-1.94 - 2.68i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (3.89 - 5.36i)T + (-7.72 - 23.7i)T^{2} \)
7 \( 1 + (-2.47 + 7.60i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (-3.23 + 2.35i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (7.79 - 10.7i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-1.85 - 5.70i)T + (-292. + 212. i)T^{2} \)
23 \( 1 - 6.63iT - 529T^{2} \)
29 \( 1 + (-37.8 - 12.2i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-21.0 + 15.2i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-9.27 + 28.5i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-12.6 + 4.09i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 42T + 1.84e3T^{2} \)
47 \( 1 + (82.0 - 26.6i)T + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-35.0 - 48.2i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (-63.0 - 20.4i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-9.70 - 7.05i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 2T + 4.48e3T^{2} \)
71 \( 1 + (35.0 - 48.2i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-22.8 + 70.3i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (32.3 - 23.5i)T + (1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-23.3 + 32.1i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 + (50.1 - 36.4i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−11.68428411747846896595914930907, −10.79864977739941511077902996781, −10.13534096635930487368673574446, −8.504455559902708514102041720316, −7.79447689023140537239583808624, −7.00232033525939463212403665757, −6.01599627571608729456957169680, −4.68845270593393339994967509620, −3.99304683954337036343731870272, −3.18160910696266606170493430349, 0.811292809446802322404575139449, 2.09523237694257881029268506584, 3.15027772003636097619838016139, 4.54047235218903439872826938531, 5.28439749330804372913582015942, 6.60818789717111402306161419510, 8.208259799721494067553046506917, 8.670198969255423563770542905710, 9.835984100587268390790404610191, 11.40847978307242292189545721495

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