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Properties

Label 2-363-33.2-c1-0-16
Degree $2$
Conductor $363$
Sign $0.996 + 0.0810i$
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  + (0.309 − 0.951i)2-s + (0.0222 + 1.73i)3-s + (0.809 + 0.587i)4-s + (2.68 − 0.874i)5-s + (1.65 + 0.514i)6-s + (0.831 − 1.14i)7-s + (2.42 − 1.76i)8-s + (−2.99 + 0.0770i)9-s − 2.82i·10-s + (−0.999 + 1.41i)12-s + (−4.03 − 1.31i)13-s + (−0.831 − 1.14i)14-s + (1.57 + 4.63i)15-s + (−0.309 − 0.951i)16-s + (1.85 + 5.70i)17-s + (−0.853 + 2.87i)18-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.0128 + 0.999i)3-s + (0.404 + 0.293i)4-s + (1.20 − 0.390i)5-s + (0.675 + 0.209i)6-s + (0.314 − 0.432i)7-s + (0.858 − 0.623i)8-s + (−0.999 + 0.0256i)9-s − 0.894i·10-s + (−0.288 + 0.408i)12-s + (−1.11 − 0.363i)13-s + (−0.222 − 0.305i)14-s + (0.406 + 1.19i)15-s + (−0.0772 − 0.237i)16-s + (0.449 + 1.38i)17-s + (−0.201 + 0.677i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.996 + 0.0810i$
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: \((2,\ 363,\ (\ :1/2),\ 0.996 + 0.0810i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01751 - 0.0819183i\)
\(L(\frac12)\) \(\approx\) \(2.01751 - 0.0819183i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0222 - 1.73i)T \)
11 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-2.68 + 0.874i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.831 + 1.14i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (4.03 + 1.31i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.85 - 5.70i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.49 + 3.43i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-1.61 - 1.17i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.618 - 1.90i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.47 + 4.70i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.85 - 3.52i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + (-1.66 - 2.28i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.37 + 1.74i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.65 + 9.15i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (9.41 - 3.05i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (-2.68 + 0.874i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.831 + 1.14i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.03 + 1.31i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.94 + 15.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (0.618 - 1.90i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.19302328226453566684042003965, −10.42407818837679867825239261468, −9.965967358418775306181395170854, −8.914961312628916849725996941718, −7.79393320955948537729590561498, −6.44022942861688013035455576075, −5.26152506050795653581544304443, −4.34410123473468676769700925314, −3.07931351541466782478857295826, −1.84998781243260543389574104742, 1.81696958320895925347014336166, 2.59914136017657930514177624396, 5.10881160582943002626847091896, 5.74680609094348696078408556130, 6.70935823732349143746829774286, 7.30085369308278548454662267675, 8.394461617200762413754662219694, 9.631849438846569324424665707655, 10.47641247784461986243103291847, 11.66723778871418743382178631410

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