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Properties

Label 2-558-31.14-c1-0-6
Degree $2$
Conductor $558$
Sign $0.799 - 0.600i$
Analytic cond. $4.45565$
Root an. cond. $2.11084$
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 + (−0.809 − 0.587i)4-s + (−0.360 + 0.623i)5-s + (0.173 − 1.65i)7-s + (0.809 − 0.587i)8-s + (−0.481 − 0.535i)10-s + (1.28 − 0.573i)11-s + (2.72 + 0.578i)13-s + (1.51 + 0.675i)14-s + (0.309 + 0.951i)16-s + (0.388 + 0.173i)17-s + (2.25 − 0.479i)19-s + (0.657 − 0.292i)20-s + (0.147 + 1.40i)22-s + (4.74 − 3.44i)23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.161 + 0.278i)5-s + (0.0656 − 0.624i)7-s + (0.286 − 0.207i)8-s + (−0.152 − 0.169i)10-s + (0.388 − 0.172i)11-s + (0.755 + 0.160i)13-s + (0.405 + 0.180i)14-s + (0.0772 + 0.237i)16-s + (0.0943 + 0.0419i)17-s + (0.517 − 0.110i)19-s + (0.147 − 0.0655i)20-s + (0.0313 + 0.298i)22-s + (0.990 − 0.719i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.799 - 0.600i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ 0.799 - 0.600i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23814 + 0.412955i\)
\(L(\frac12)\) \(\approx\) \(1.23814 + 0.412955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 \)
31 \( 1 + (-4.91 - 2.60i)T \)
good5 \( 1 + (0.360 - 0.623i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.173 + 1.65i)T + (-6.84 - 1.45i)T^{2} \)
11 \( 1 + (-1.28 + 0.573i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (-2.72 - 0.578i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (-0.388 - 0.173i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (-2.25 + 0.479i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (-4.74 + 3.44i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.0752 - 0.231i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (-1.38 - 2.39i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.38 + 3.75i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (2.77 - 0.590i)T + (39.2 - 17.4i)T^{2} \)
47 \( 1 + (-2.12 - 6.55i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.501 + 4.77i)T + (-51.8 + 11.0i)T^{2} \)
59 \( 1 + (-8.08 + 8.97i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + 2.61T + 61T^{2} \)
67 \( 1 + (-2.62 + 4.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.12 - 10.7i)T + (-69.4 + 14.7i)T^{2} \)
73 \( 1 + (5.42 - 2.41i)T + (48.8 - 54.2i)T^{2} \)
79 \( 1 + (12.2 + 5.44i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (-4.01 - 4.45i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + (-0.481 - 0.350i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (4.23 + 3.07i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.81814146111737123497091419805, −9.923030998324676209973020711293, −8.937094610631663955817343967720, −8.198710548104854815665737126745, −7.11281746059834246302630836387, −6.58334249282610543857318912213, −5.39374130051011808820881592865, −4.30326911423235563123763377111, −3.18205586120900385808927958609, −1.12753313848498159844504944282, 1.17208825849658489999818833742, 2.66262683780225003017299007732, 3.81141499710637484303701542503, 4.94608924981851949850735503537, 5.98448611653426154154558141357, 7.21512651202568926499034757482, 8.333032422603810296261458025328, 8.935342193032267059594115871924, 9.807237274551227866102133906544, 10.71521289901764800837061638849

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