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Properties

Label 2-558-279.211-c1-0-18
Degree $2$
Conductor $558$
Sign $0.963 + 0.267i$
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  + (−0.5 − 0.866i)2-s + (1.72 − 0.201i)3-s + (−0.499 + 0.866i)4-s + (2.15 + 3.73i)5-s + (−1.03 − 1.38i)6-s + (1.90 − 3.29i)7-s + 0.999·8-s + (2.91 − 0.692i)9-s + (2.15 − 3.73i)10-s − 3.83·11-s + (−0.685 + 1.59i)12-s + (0.110 + 0.191i)13-s − 3.80·14-s + (4.45 + 5.98i)15-s + (−0.5 − 0.866i)16-s + (1.50 + 2.61i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.993 − 0.116i)3-s + (−0.249 + 0.433i)4-s + (0.963 + 1.66i)5-s + (−0.422 − 0.567i)6-s + (0.718 − 1.24i)7-s + 0.353·8-s + (0.972 − 0.230i)9-s + (0.681 − 1.17i)10-s − 1.15·11-s + (−0.197 + 0.459i)12-s + (0.0306 + 0.0531i)13-s − 1.01·14-s + (1.15 + 1.54i)15-s + (−0.125 − 0.216i)16-s + (0.365 + 0.633i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.963 + 0.267i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ 0.963 + 0.267i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99004 - 0.270664i\)
\(L(\frac12)\) \(\approx\) \(1.99004 - 0.270664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1.72 + 0.201i)T \)
31 \( 1 + (0.177 - 5.56i)T \)
good5 \( 1 + (-2.15 - 3.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.90 + 3.29i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.83T + 11T^{2} \)
13 \( 1 + (-0.110 - 0.191i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.50 - 2.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.48 + 2.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.54 + 4.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.95 - 5.11i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (4.49 + 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.15 - 7.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.155 - 0.268i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.06 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.763 + 1.32i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 + (-0.797 - 1.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.88 + 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.12 - 3.67i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.91 + 6.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.59 - 7.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.99T + 83T^{2} \)
89 \( 1 - 8.12T + 89T^{2} \)
97 \( 1 + (3.03 + 5.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.49625592871826261351624263014, −10.27168070034068322718140771031, −9.095578714895472489803651712719, −8.022109250849011624338916673549, −7.28731571110943789766277911337, −6.58460240217742036743206807874, −4.87734179153510958988850708553, −3.51584859008728120579436832430, −2.71661896564942698681085646692, −1.68255116869044731074243966328, 1.50392778520655460034813439790, 2.54580137834363209095905960418, 4.58477719614464263686950603957, 5.24967083705816753340121144327, 5.98046088308775561390842190272, 7.78473820740942594616789254119, 8.189610106367177696342333712878, 9.022090566397900753093801052175, 9.488619406298328760245539184434, 10.33241940477505362604123106376

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