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Properties

Label 2-363-33.2-c1-0-18
Degree $2$
Conductor $363$
Sign $0.998 - 0.0515i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.465 + 1.43i)2-s + (0.479 − 1.66i)3-s + (−0.216 − 0.157i)4-s + (2.76 − 0.899i)5-s + (2.16 + 1.46i)6-s + (1.66 − 2.28i)7-s + (−2.11 + 1.53i)8-s + (−2.54 − 1.59i)9-s + 4.38i·10-s + (−0.366 + 0.285i)12-s + (0.852 + 0.277i)13-s + (2.50 + 3.44i)14-s + (−0.170 − 5.03i)15-s + (−1.37 − 4.24i)16-s + (−0.806 − 2.48i)17-s + (3.46 − 2.89i)18-s + ⋯
L(s)  = 1  + (−0.329 + 1.01i)2-s + (0.276 − 0.960i)3-s + (−0.108 − 0.0787i)4-s + (1.23 − 0.402i)5-s + (0.882 + 0.596i)6-s + (0.628 − 0.864i)7-s + (−0.746 + 0.542i)8-s + (−0.846 − 0.531i)9-s + 1.38i·10-s + (−0.105 + 0.0823i)12-s + (0.236 + 0.0768i)13-s + (0.669 + 0.920i)14-s + (−0.0440 − 1.30i)15-s + (−0.344 − 1.06i)16-s + (−0.195 − 0.601i)17-s + (0.817 − 0.682i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.998 - 0.0515i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.998 - 0.0515i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58954 + 0.0410074i\)
\(L(\frac12)\) \(\approx\) \(1.58954 + 0.0410074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.479 + 1.66i)T \)
11 \( 1 \)
good2 \( 1 + (0.465 - 1.43i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-2.76 + 0.899i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.66 + 2.28i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.852 - 0.277i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.806 + 2.48i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.43 + 1.98i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-6.98 - 5.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.15 - 9.69i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.86 - 2.07i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.65 + 2.65i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + (1.25 + 1.72i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.27 - 2.03i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.335 + 0.461i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (9.67 - 3.14i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (9.04 - 2.93i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.88 - 2.59i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (6.98 + 2.27i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.95 + 6.00i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 + (-1.77 + 5.45i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.49803073273087374241995107281, −10.42566725299159123702338701025, −9.076654198321291671352412010003, −8.593144725475623417248156891104, −7.48156012695390195740214916633, −6.83041947997257066951851796703, −5.96399651405033447047457932996, −4.91064493559284962137359102537, −2.80279967391852704277837537269, −1.37599976048625770926527777172, 1.99306174869487159614214100175, 2.68938166164620366053350837059, 4.13800370788835162547465012512, 5.65122688236062008087214075512, 6.20582139723164146104038159092, 8.158537431135453733517287891323, 9.079574638062816344577896065816, 9.774607356459494760205239969063, 10.43783561762609563971097955746, 11.16011146568405230868009578564

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