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Properties

Label 2-688-43.38-c1-0-17
Degree $2$
Conductor $688$
Sign $-0.921 + 0.389i$
Analytic cond. $5.49370$
Root an. cond. $2.34386$
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  + (0.200 + 0.0618i)3-s + (0.961 + 2.44i)5-s + (−1.57 − 2.72i)7-s + (−2.44 − 1.66i)9-s + (−4.94 + 2.37i)11-s + (−6.34 − 0.956i)13-s + (0.0412 + 0.550i)15-s + (1.97 − 5.04i)17-s + (−2.08 + 1.42i)19-s + (−0.147 − 0.644i)21-s + (0.168 − 2.24i)23-s + (−1.40 + 1.30i)25-s + (−0.779 − 0.977i)27-s + (1.23 − 0.380i)29-s + (5.16 + 4.79i)31-s + ⋯
L(s)  = 1  + (0.115 + 0.0357i)3-s + (0.429 + 1.09i)5-s + (−0.595 − 1.03i)7-s + (−0.814 − 0.555i)9-s + (−1.48 + 0.717i)11-s + (−1.75 − 0.265i)13-s + (0.0106 + 0.142i)15-s + (0.480 − 1.22i)17-s + (−0.478 + 0.325i)19-s + (−0.0320 − 0.140i)21-s + (0.0350 − 0.467i)23-s + (−0.281 + 0.261i)25-s + (−0.149 − 0.188i)27-s + (0.228 − 0.0706i)29-s + (0.928 + 0.861i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(688\)    =    \(2^{4} \cdot 43\)
Sign: $-0.921 + 0.389i$
Analytic conductor: \(5.49370\)
Root analytic conductor: \(2.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{688} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 688,\ (\ :1/2),\ -0.921 + 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0311824 - 0.153828i\)
\(L(\frac12)\) \(\approx\) \(0.0311824 - 0.153828i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 + (2.14 + 6.19i)T \)
good3 \( 1 + (-0.200 - 0.0618i)T + (2.47 + 1.68i)T^{2} \)
5 \( 1 + (-0.961 - 2.44i)T + (-3.66 + 3.40i)T^{2} \)
7 \( 1 + (1.57 + 2.72i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.94 - 2.37i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (6.34 + 0.956i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (-1.97 + 5.04i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (2.08 - 1.42i)T + (6.94 - 17.6i)T^{2} \)
23 \( 1 + (-0.168 + 2.24i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (-1.23 + 0.380i)T + (23.9 - 16.3i)T^{2} \)
31 \( 1 + (-5.16 - 4.79i)T + (2.31 + 30.9i)T^{2} \)
37 \( 1 + (3.36 - 5.82i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.120 - 0.528i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-1.23 - 0.596i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-2.70 + 0.407i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-0.595 - 0.746i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (7.10 - 6.59i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (5.68 - 3.87i)T + (24.4 - 62.3i)T^{2} \)
71 \( 1 + (-0.107 - 1.43i)T + (-70.2 + 10.5i)T^{2} \)
73 \( 1 + (8.69 + 1.31i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-5.03 - 8.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.04 + 2.17i)T + (68.5 + 46.7i)T^{2} \)
89 \( 1 + (12.9 + 4.00i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 + (1.92 - 0.925i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.27584162307629837685682595079, −9.604222535940265231015486446709, −8.222496923903923545967025492828, −7.16646984664486488625323031897, −6.89298687406519533842348780557, −5.57677612678338007053211393647, −4.60247248608402786720251646606, −2.98824897982246325567392747876, −2.64326383922091777918423993843, −0.07220740607817684005484757176, 2.16921205331196727374041488404, 2.94958784567584274695227944452, 4.75382349361454498870296111801, 5.46223817870885832334477307133, 6.05089004907112946087011575009, 7.64074795961385266799624950393, 8.357195668178941596894777989368, 9.052696876456399207546570665409, 9.872209036258065239156518173019, 10.73482181231155734097136211426

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