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Properties

Label 2-688-43.40-c1-0-0
Degree $2$
Conductor $688$
Sign $-0.548 - 0.835i$
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  + (0.0482 − 0.644i)3-s + (−3.17 − 0.979i)5-s + (−1.23 − 2.13i)7-s + (2.55 + 0.384i)9-s + (−0.748 + 0.937i)11-s + (−4.22 + 3.91i)13-s + (−0.784 + 1.99i)15-s + (1.02 − 0.314i)17-s + (−2.13 + 0.321i)19-s + (−1.43 + 0.691i)21-s + (2.16 + 5.50i)23-s + (4.98 + 3.40i)25-s + (0.802 − 3.51i)27-s + (0.177 + 2.37i)29-s + (−6.31 + 4.30i)31-s + ⋯
L(s)  = 1  + (0.0278 − 0.371i)3-s + (−1.41 − 0.437i)5-s + (−0.465 − 0.807i)7-s + (0.851 + 0.128i)9-s + (−0.225 + 0.282i)11-s + (−1.17 + 1.08i)13-s + (−0.202 + 0.515i)15-s + (0.247 − 0.0763i)17-s + (−0.489 + 0.0737i)19-s + (−0.313 + 0.150i)21-s + (0.450 + 1.14i)23-s + (0.997 + 0.680i)25-s + (0.154 − 0.676i)27-s + (0.0330 + 0.440i)29-s + (−1.13 + 0.773i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(688\)    =    \(2^{4} \cdot 43\)
Sign: $-0.548 - 0.835i$
Analytic conductor: \(5.49370\)
Root analytic conductor: \(2.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{688} (513, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 688,\ (\ :1/2),\ -0.548 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102877 + 0.190648i\)
\(L(\frac12)\) \(\approx\) \(0.102877 + 0.190648i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 - 6.08i)T \)
good3 \( 1 + (-0.0482 + 0.644i)T + (-2.96 - 0.447i)T^{2} \)
5 \( 1 + (3.17 + 0.979i)T + (4.13 + 2.81i)T^{2} \)
7 \( 1 + (1.23 + 2.13i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.748 - 0.937i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (4.22 - 3.91i)T + (0.971 - 12.9i)T^{2} \)
17 \( 1 + (-1.02 + 0.314i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (2.13 - 0.321i)T + (18.1 - 5.60i)T^{2} \)
23 \( 1 + (-2.16 - 5.50i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (-0.177 - 2.37i)T + (-28.6 + 4.32i)T^{2} \)
31 \( 1 + (6.31 - 4.30i)T + (11.3 - 28.8i)T^{2} \)
37 \( 1 + (2.83 - 4.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.77 + 4.22i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (3.30 + 4.14i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-2.69 - 2.50i)T + (3.96 + 52.8i)T^{2} \)
59 \( 1 + (0.208 - 0.914i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (2.31 + 1.57i)T + (22.2 + 56.7i)T^{2} \)
67 \( 1 + (11.8 - 1.78i)T + (64.0 - 19.7i)T^{2} \)
71 \( 1 + (-5.54 + 14.1i)T + (-52.0 - 48.2i)T^{2} \)
73 \( 1 + (4.99 - 4.63i)T + (5.45 - 72.7i)T^{2} \)
79 \( 1 + (-1.18 - 2.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.533 + 7.11i)T + (-82.0 - 12.3i)T^{2} \)
89 \( 1 + (0.113 - 1.51i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (6.73 - 8.44i)T + (-21.5 - 94.5i)T^{2} \)
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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

−10.76084082644097607530084925585, −9.927651595841110023767621388698, −9.028677388787552445630095137870, −7.922691021052373047502681269292, −7.18211362520128858961176474636, −6.88914307595758447093684575564, −5.03611024781630786152568490985, −4.29015091753042440365997138787, −3.40688121749887258968127744684, −1.59956814973107627129325877756, 0.11369272077022555720196263529, 2.59861114041068066140443984060, 3.54500668176523908641089114764, 4.51427378630780408660830241195, 5.56230097967969539151309970249, 6.83221934652900201021368621834, 7.56595900473738468514037990433, 8.366786750509963918894118251155, 9.359683416502425693187172179868, 10.29767530057459000958135760312

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