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Properties

Label 2-688-43.15-c1-0-1
Degree $2$
Conductor $688$
Sign $-0.929 - 0.368i$
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  + (−0.922 + 2.34i)3-s + (−0.0373 − 0.498i)5-s + (1.65 − 2.86i)7-s + (−2.47 − 2.29i)9-s + (−0.828 + 3.62i)11-s + (−3.87 + 2.64i)13-s + (1.20 + 0.371i)15-s + (−0.343 + 4.57i)17-s + (−4.44 + 4.12i)19-s + (5.20 + 6.52i)21-s + (0.371 − 0.114i)23-s + (4.69 − 0.707i)25-s + (0.844 − 0.406i)27-s + (−1.29 − 3.29i)29-s + (−0.861 − 0.129i)31-s + ⋯
L(s)  = 1  + (−0.532 + 1.35i)3-s + (−0.0167 − 0.222i)5-s + (0.624 − 1.08i)7-s + (−0.823 − 0.764i)9-s + (−0.249 + 1.09i)11-s + (−1.07 + 0.732i)13-s + (0.311 + 0.0960i)15-s + (−0.0831 + 1.11i)17-s + (−1.01 + 0.945i)19-s + (1.13 + 1.42i)21-s + (0.0774 − 0.0239i)23-s + (0.939 − 0.141i)25-s + (0.162 − 0.0782i)27-s + (−0.239 − 0.611i)29-s + (−0.154 − 0.0233i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(688\)    =    \(2^{4} \cdot 43\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(5.49370\)
Root analytic conductor: \(2.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{688} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 688,\ (\ :1/2),\ -0.929 - 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.150369 + 0.787228i\)
\(L(\frac12)\) \(\approx\) \(0.150369 + 0.787228i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.26 - 1.92i)T \)
good3 \( 1 + (0.922 - 2.34i)T + (-2.19 - 2.04i)T^{2} \)
5 \( 1 + (0.0373 + 0.498i)T + (-4.94 + 0.745i)T^{2} \)
7 \( 1 + (-1.65 + 2.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.828 - 3.62i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (3.87 - 2.64i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (0.343 - 4.57i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (4.44 - 4.12i)T + (1.41 - 18.9i)T^{2} \)
23 \( 1 + (-0.371 + 0.114i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (1.29 + 3.29i)T + (-21.2 + 19.7i)T^{2} \)
31 \( 1 + (0.861 + 0.129i)T + (29.6 + 9.13i)T^{2} \)
37 \( 1 + (1.96 + 3.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.72 - 8.43i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-0.776 - 3.40i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-3.94 - 2.68i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (-6.05 + 2.91i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (5.18 - 0.782i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (-7.73 + 7.17i)T + (5.00 - 66.8i)T^{2} \)
71 \( 1 + (0.859 + 0.265i)T + (58.6 + 39.9i)T^{2} \)
73 \( 1 + (0.798 - 0.544i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (0.500 - 0.867i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.24 + 3.17i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (0.672 - 1.71i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (0.213 - 0.936i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.54605284169379134485884028481, −10.21298240810839076662282069160, −9.445433412220194966774740653147, −8.314039152685949487056372045607, −7.37857000115952614590739691827, −6.34167382190566313985557768291, −4.98590808867912518731817344185, −4.51797111450090735005708147262, −3.81306901452704406869194812424, −1.88072702530107148120390976402, 0.43371372739521510166444830271, 2.09861199763177428618441251952, 2.97226809665877338755185140394, 5.07462567705750560830023877695, 5.51872039262792846421345752666, 6.71562927140129182906110274185, 7.25864410982049954848494135501, 8.362854545343141125525102421045, 8.881639059116453417943871830246, 10.30453109527514912936110503326

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