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Properties

Label 2-588-12.11-c1-0-29
Degree $2$
Conductor $588$
Sign $-0.167 - 0.985i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
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  + (0.866 + 1.11i)2-s + (1.58 − 0.707i)3-s + (−0.500 + 1.93i)4-s + 2.44i·5-s + (2.15 + 1.15i)6-s + (−2.59 + 1.11i)8-s + (2.00 − 2.23i)9-s + (−2.73 + 2.12i)10-s − 3.46·11-s + (0.578 + 3.41i)12-s + 5.47·13-s + (1.73 + 3.87i)15-s + (−3.5 − 1.93i)16-s + 4.89i·17-s + (4.23 + 0.299i)18-s + 4.24i·19-s + ⋯
L(s)  = 1  + (0.612 + 0.790i)2-s + (0.912 − 0.408i)3-s + (−0.250 + 0.968i)4-s + 1.09i·5-s + (0.881 + 0.471i)6-s + (−0.918 + 0.395i)8-s + (0.666 − 0.745i)9-s + (−0.866 + 0.670i)10-s − 1.04·11-s + (0.167 + 0.985i)12-s + 1.51·13-s + (0.447 + 0.999i)15-s + (−0.875 − 0.484i)16-s + 1.18i·17-s + (0.997 + 0.0706i)18-s + 0.973i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.167 - 0.985i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.167 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64343 + 1.94533i\)
\(L(\frac12)\) \(\approx\) \(1.64343 + 1.94533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 1.11i)T \)
3 \( 1 + (-1.58 + 0.707i)T \)
7 \( 1 \)
good5 \( 1 - 2.44iT - 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - 5.47T + 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 7.74iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 4.47iT - 53T^{2} \)
59 \( 1 - 9.48T + 59T^{2} \)
61 \( 1 + 5.47T + 61T^{2} \)
67 \( 1 - 7.74iT - 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 - 9.48T + 83T^{2} \)
89 \( 1 + 9.79iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.89698667431357007637453399295, −10.06609333626109019824795417476, −8.759693672222205916176832641483, −8.069441200465467101735866936943, −7.44125819607794697603139724143, −6.37700479042904094690756629941, −5.82144924215542053459077455220, −4.04461368226580306873781377094, −3.38951714953585904657040784067, −2.24682076821343114350555193694, 1.20357262065163400285475578150, 2.66859900605976416864466954447, 3.61275881453714898501354785219, 4.85191607807871687290837366494, 5.20877664513282737465349587097, 6.79154545634358279785240810705, 8.200280164027261268624317960389, 8.895209810318371171849152710460, 9.500484113621841943629521758088, 10.58352606087912046590881495087

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