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Properties

Label 2-558-31.25-c1-0-4
Degree $2$
Conductor $558$
Sign $0.275 - 0.961i$
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  + 2-s + 4-s + (−1.5 + 2.59i)5-s + (0.5 + 0.866i)7-s + 8-s + (−1.5 + 2.59i)10-s + (1.5 − 2.59i)11-s + (−2.5 + 4.33i)13-s + (0.5 + 0.866i)14-s + 16-s + (1.5 + 2.59i)17-s + (3.5 + 6.06i)19-s + (−1.5 + 2.59i)20-s + (1.5 − 2.59i)22-s + (−2 − 3.46i)25-s + (−2.5 + 4.33i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.670 + 1.16i)5-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.474 + 0.821i)10-s + (0.452 − 0.783i)11-s + (−0.693 + 1.20i)13-s + (0.133 + 0.231i)14-s + 0.250·16-s + (0.363 + 0.630i)17-s + (0.802 + 1.39i)19-s + (−0.335 + 0.580i)20-s + (0.319 − 0.553i)22-s + (−0.400 − 0.692i)25-s + (−0.490 + 0.849i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ 0.275 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55758 + 1.17446i\)
\(L(\frac12)\) \(\approx\) \(1.55758 + 1.17446i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
31 \( 1 + (2 + 5.19i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−11.22275943386260477331696097932, −10.30030455167212410108549701495, −9.268378053049533752810937914799, −7.998721917769142684388514085320, −7.27294586570844070070191643806, −6.34870032885874448028992261736, −5.49015616129293459542646032424, −4.01838039073801222850824626673, −3.40897137906598837640320817197, −2.03662367753396185886817406214, 0.949734282541615877198605666525, 2.78245278083099687735775899975, 4.12113625375092048794348631041, 4.87603865224511495515451097049, 5.60778260663906808050098259181, 7.30508308588942493336678610469, 7.53534120707006903524393337716, 8.865049762419181923665004600083, 9.645577726413583026226755782301, 10.79993670702986353973431177711

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