Properties

Label 2-688-43.42-c2-0-28
Degree $2$
Conductor $688$
Sign $0.950 + 0.309i$
Analytic cond. $18.7466$
Root an. cond. $4.32973$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.61i·3-s − 2.98i·5-s − 6.83i·7-s − 22.5·9-s − 6.88·11-s + 12.4·13-s + 16.7·15-s − 15.8·17-s − 19.2i·19-s + 38.3·21-s + 33.2·23-s + 16.0·25-s − 76.1i·27-s − 45.8i·29-s − 14.7·31-s + ⋯
L(s)  = 1  + 1.87i·3-s − 0.597i·5-s − 0.975i·7-s − 2.50·9-s − 0.626·11-s + 0.953·13-s + 1.11·15-s − 0.930·17-s − 1.01i·19-s + 1.82·21-s + 1.44·23-s + 0.642·25-s − 2.82i·27-s − 1.58i·29-s − 0.475·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(688\)    =    \(2^{4} \cdot 43\)
Sign: $0.950 + 0.309i$
Analytic conductor: \(18.7466\)
Root analytic conductor: \(4.32973\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{688} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 688,\ (\ :1),\ 0.950 + 0.309i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.329556858\)
\(L(\frac12)\) \(\approx\) \(1.329556858\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 + (-40.8 - 13.3i)T \)
good3 \( 1 - 5.61iT - 9T^{2} \)
5 \( 1 + 2.98iT - 25T^{2} \)
7 \( 1 + 6.83iT - 49T^{2} \)
11 \( 1 + 6.88T + 121T^{2} \)
13 \( 1 - 12.4T + 169T^{2} \)
17 \( 1 + 15.8T + 289T^{2} \)
19 \( 1 + 19.2iT - 361T^{2} \)
23 \( 1 - 33.2T + 529T^{2} \)
29 \( 1 + 45.8iT - 841T^{2} \)
31 \( 1 + 14.7T + 961T^{2} \)
37 \( 1 + 13.1iT - 1.36e3T^{2} \)
41 \( 1 + 2.49T + 1.68e3T^{2} \)
47 \( 1 - 10.2T + 2.20e3T^{2} \)
53 \( 1 + 31.2T + 2.80e3T^{2} \)
59 \( 1 + 64.4T + 3.48e3T^{2} \)
61 \( 1 + 78.1iT - 3.72e3T^{2} \)
67 \( 1 - 89.3T + 4.48e3T^{2} \)
71 \( 1 + 35.5iT - 5.04e3T^{2} \)
73 \( 1 - 35.9iT - 5.32e3T^{2} \)
79 \( 1 + 50.1T + 6.24e3T^{2} \)
83 \( 1 - 10.3T + 6.88e3T^{2} \)
89 \( 1 + 13.4iT - 7.92e3T^{2} \)
97 \( 1 - 66.6T + 9.40e3T^{2} \)
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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

−10.32090820696979235142580552288, −9.266485595984761778635236092674, −8.913441070511483958739195516218, −7.87030289996590869151583384534, −6.51758090578239721745847638525, −5.31178635634272454785909645984, −4.59931543981958564065294772380, −3.93298601390195353200186982962, −2.81158146645241148601081952703, −0.51218581229114630577449401835, 1.27140451336358858423526151304, 2.39298100985171063675837955970, 3.20913642521113627050790991176, 5.26636239408128263515256631613, 6.08330383427982316352246653651, 6.81882761737854906327382211924, 7.53444668517534458856338928020, 8.570574044189909799332650334224, 8.963035020878450627930505372526, 10.73393736472804450694621443480

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