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Properties

Label 2-588-28.27-c1-0-26
Degree $2$
Conductor $588$
Sign $-0.704 + 0.709i$
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  + (0.639 − 1.26i)2-s − 3-s + (−1.18 − 1.61i)4-s + 3.10i·5-s + (−0.639 + 1.26i)6-s + (−2.79 + 0.457i)8-s + 9-s + (3.91 + 1.98i)10-s − 5.34i·11-s + (1.18 + 1.61i)12-s − 3.92i·13-s − 3.10i·15-s + (−1.20 + 3.81i)16-s − 5.68i·17-s + (0.639 − 1.26i)18-s − 0.170·19-s + ⋯
L(s)  = 1  + (0.452 − 0.891i)2-s − 0.577·3-s + (−0.590 − 0.806i)4-s + 1.38i·5-s + (−0.261 + 0.514i)6-s + (−0.986 + 0.161i)8-s + 0.333·9-s + (1.23 + 0.628i)10-s − 1.61i·11-s + (0.340 + 0.465i)12-s − 1.08i·13-s − 0.801i·15-s + (−0.302 + 0.953i)16-s − 1.37i·17-s + (0.150 − 0.297i)18-s − 0.0391·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.704 + 0.709i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430084 - 1.03304i\)
\(L(\frac12)\) \(\approx\) \(0.430084 - 1.03304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.639 + 1.26i)T \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 3.10iT - 5T^{2} \)
11 \( 1 + 5.34iT - 11T^{2} \)
13 \( 1 + 3.92iT - 13T^{2} \)
17 \( 1 + 5.68iT - 17T^{2} \)
19 \( 1 + 0.170T + 19T^{2} \)
23 \( 1 + 6.13iT - 23T^{2} \)
29 \( 1 + 1.96T + 29T^{2} \)
31 \( 1 - 1.53T + 31T^{2} \)
37 \( 1 - 8.93T + 37T^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 + 2.38iT - 43T^{2} \)
47 \( 1 + 2.29T + 47T^{2} \)
53 \( 1 + 1.19T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 6.58iT - 61T^{2} \)
67 \( 1 - 9.64iT - 67T^{2} \)
71 \( 1 + 2.04iT - 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 5.81iT - 89T^{2} \)
97 \( 1 + 2.32iT - 97T^{2} \)
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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

−10.60549165118473811409512164793, −9.980208861905069108042470604726, −8.812020518639266538853757483561, −7.61210893216276021140889715594, −6.36273577349597030088216390654, −5.83528969026161052729036223945, −4.66279053565537984740105968125, −3.24716173368295718757460436745, −2.72848621006591660369925944486, −0.60076418645101450947073223526, 1.69219235016130061996777563706, 4.03531849206192405379490504703, 4.58636719009459178099324451771, 5.45492010373388648631169056549, 6.40818424272777509812783397556, 7.38169986699296504760537881130, 8.223746058600037526805092382024, 9.270219114265315504440344477012, 9.753638936969314909320811628960, 11.29178456672630208027775100545

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