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Properties

Label 2-363-33.8-c1-0-5
Degree $2$
Conductor $363$
Sign $-0.237 - 0.971i$
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  + (−0.951 + 0.690i)2-s + (−1.72 − 0.0877i)3-s + (−0.190 + 0.587i)4-s + (0.224 − 0.309i)5-s + (1.70 − 1.11i)6-s + (2.92 + 0.951i)7-s + (−0.951 − 2.92i)8-s + (2.98 + 0.303i)9-s + 0.449i·10-s + (0.381 − i)12-s + (−0.427 − 0.587i)13-s + (−3.44 + 1.11i)14-s + (−0.415 + 0.514i)15-s + (1.92 + 1.40i)16-s + (3.44 + 2.5i)17-s + (−3.04 + 1.77i)18-s + ⋯
L(s)  = 1  + (−0.672 + 0.488i)2-s + (−0.998 − 0.0506i)3-s + (−0.0954 + 0.293i)4-s + (0.100 − 0.138i)5-s + (0.696 − 0.453i)6-s + (1.10 + 0.359i)7-s + (−0.336 − 1.03i)8-s + (0.994 + 0.101i)9-s + 0.141i·10-s + (0.110 − 0.288i)12-s + (−0.118 − 0.163i)13-s + (−0.919 + 0.298i)14-s + (−0.107 + 0.132i)15-s + (0.481 + 0.350i)16-s + (0.834 + 0.606i)17-s + (−0.718 + 0.418i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.237 - 0.971i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.237 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.416919 + 0.531074i\)
\(L(\frac12)\) \(\approx\) \(0.416919 + 0.531074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.0877i)T \)
11 \( 1 \)
good2 \( 1 + (0.951 - 0.690i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.224 + 0.309i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-2.92 - 0.951i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.427 + 0.587i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.44 - 2.5i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.5 - 0.812i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.23iT - 23T^{2} \)
29 \( 1 + (0.726 - 2.23i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.73 - 3.44i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.927 + 2.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.224 - 0.690i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.88iT - 43T^{2} \)
47 \( 1 + (-7.91 + 2.57i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.08 + 1.5i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.363 - 0.118i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.54 - 2.12i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.32T + 67T^{2} \)
71 \( 1 + (3.13 - 4.30i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-8.94 - 2.90i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.33 + 3.21i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.11 + 3.71i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.527iT - 89T^{2} \)
97 \( 1 + (-3.54 + 2.57i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.62271068436719380704215219198, −10.80288387739843746322547874747, −9.782584320088643290383506292824, −8.822950949862713728197360958818, −7.83838677627478830896823852166, −7.17964590900325707094562551684, −5.90365751504139223085873102749, −5.06661595328533012418479376994, −3.74391992654844642399222668308, −1.45037382437452631066300191552, 0.73383767677825996304746693894, 2.15779757305991594503529318329, 4.37329514535236683122641015280, 5.21282738843760125722728911617, 6.25099140685583179562877028439, 7.47291757749245754462518592994, 8.496159969019539943974716505192, 9.611726558991758681260263900859, 10.46749598942005205931846794623, 10.96111927004153088232197364953

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