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Properties

Label 2-363-11.3-c1-0-17
Degree $2$
Conductor $363$
Sign $-0.927 + 0.374i$
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  + (1.61 − 1.17i)2-s + (−0.309 − 0.951i)3-s + (0.618 − 1.90i)4-s + (−3.23 − 2.35i)5-s + (−1.61 − 1.17i)6-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)9-s − 8·10-s − 2.00·12-s + (1.61 − 1.17i)13-s + (−0.618 − 1.90i)14-s + (−1.23 + 3.80i)15-s + (3.23 + 2.35i)16-s + (−3.23 − 2.35i)17-s + (−0.618 + 1.90i)18-s + (−0.927 − 2.85i)19-s + ⋯
L(s)  = 1  + (1.14 − 0.831i)2-s + (−0.178 − 0.549i)3-s + (0.309 − 0.951i)4-s + (−1.44 − 1.05i)5-s + (−0.660 − 0.479i)6-s + (0.116 − 0.359i)7-s + (−0.269 + 0.195i)9-s − 2.52·10-s − 0.577·12-s + (0.448 − 0.326i)13-s + (−0.165 − 0.508i)14-s + (−0.319 + 0.982i)15-s + (0.809 + 0.587i)16-s + (−0.784 − 0.570i)17-s + (−0.145 + 0.448i)18-s + (−0.212 − 0.654i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.927 + 0.374i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.322131 - 1.65846i\)
\(L(\frac12)\) \(\approx\) \(0.322131 - 1.65846i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-1.61 + 1.17i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (3.23 + 2.35i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.61 + 1.17i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.23 + 2.35i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.927 + 2.85i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + (-1.85 + 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.04 + 2.93i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.927 + 2.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.618 + 1.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (-0.618 - 1.90i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.85 - 3.52i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.09 - 9.51i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.42 + 1.76i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.39 - 10.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.89 - 6.46i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.85 + 3.52i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (4.04 - 2.93i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.37896408886261441260111258555, −10.76764768018248699985338455082, −9.047948435097816093092014505107, −8.128884021528328490221600093320, −7.30393750512752986916598454666, −5.81396678845320349125158428825, −4.59065229239504166699727938150, −4.14403782812745651104495231310, −2.72506616924827748787200966047, −0.872597404376471983115521268714, 3.12504292312340698071910735657, 3.97961642880742711257925137030, 4.77743798696406079333418042085, 6.12776906322719890434001836234, 6.81181152174878746713040403945, 7.79657745460773506535140235869, 8.772710341791406750767467751219, 10.33393897882517166315596108053, 11.04885348134601243696081674513, 11.92353942947076934449411595522

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