↑

Properties

Label 2-558-279.103-c1-0-22
Degree $2$
Conductor $558$
Sign $-0.0540 + 0.998i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.104 − 0.994i)2-s + (1.65 − 0.513i)3-s + (−0.978 − 0.207i)4-s − 0.102·5-s + (−0.338 − 1.69i)6-s + (−0.417 − 1.28i)7-s + (−0.309 + 0.951i)8-s + (2.47 − 1.70i)9-s + (−0.0106 + 0.101i)10-s + (4.15 + 0.884i)11-s + (−1.72 + 0.158i)12-s + (2.25 − 1.63i)13-s + (−1.32 + 0.281i)14-s + (−0.169 + 0.0526i)15-s + (0.913 + 0.406i)16-s + (−0.205 − 0.228i)17-s + ⋯
L(s)  = 1  + (0.0739 − 0.703i)2-s + (0.954 − 0.296i)3-s + (−0.489 − 0.103i)4-s − 0.0457·5-s + (−0.138 − 0.693i)6-s + (−0.157 − 0.485i)7-s + (−0.109 + 0.336i)8-s + (0.823 − 0.566i)9-s + (−0.00338 + 0.0321i)10-s + (1.25 + 0.266i)11-s + (−0.497 + 0.0458i)12-s + (0.625 − 0.454i)13-s + (−0.353 + 0.0751i)14-s + (−0.0437 + 0.0135i)15-s + (0.228 + 0.101i)16-s + (−0.0498 − 0.0553i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0540 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0540 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $-0.0540 + 0.998i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ -0.0540 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39432 - 1.47187i\)
\(L(\frac12)\) \(\approx\) \(1.39432 - 1.47187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.104 + 0.994i)T \)
3 \( 1 + (-1.65 + 0.513i)T \)
31 \( 1 + (-2.81 - 4.80i)T \)
good5 \( 1 + 0.102T + 5T^{2} \)
7 \( 1 + (0.417 + 1.28i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-4.15 - 0.884i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-2.25 + 1.63i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.205 + 0.228i)T + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (2.08 - 0.929i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (4.13 + 4.59i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (-0.963 + 9.16i)T + (-28.3 - 6.02i)T^{2} \)
37 \( 1 + (5.95 - 10.3i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.36 - 3.16i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (2.67 + 1.94i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-2.20 - 0.981i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (1.84 - 0.392i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (-8.54 - 3.80i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-7.33 - 12.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 7.57T + 67T^{2} \)
71 \( 1 + (8.59 - 1.82i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (8.68 - 9.64i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (2.41 - 7.43i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.67 + 1.19i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (1.84 + 5.68i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.14 - 5.71i)T + (-10.1 - 96.4i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.21020003237026828655659774413, −9.913327789809461976869737650576, −8.695369837182232064445229910263, −8.207676864065546562710416123190, −6.97073928786421782803974220873, −6.09587749937140996228516819291, −4.28980685427708325859182499160, −3.78362287506626610623010172942, −2.48570297460808223199075235151, −1.20220240635787587623263709577, 1.87561295652461507314710182403, 3.54182771622649101649627073950, 4.16227985308319941005127307543, 5.56051145123512852002244538593, 6.53452487130441633829864702902, 7.46466127308354288641094171304, 8.492613877902463269024400601567, 9.063502960241793149373077528827, 9.701999759024121274119214862471, 10.90932123142834538005202452404

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