Properties

Label 2-688-43.24-c1-0-0
Degree $2$
Conductor $688$
Sign $-0.703 + 0.710i$
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  + (−2.66 + 0.402i)3-s + (−2.09 + 1.43i)5-s + (1.09 + 1.89i)7-s + (4.09 − 1.26i)9-s + (−0.694 + 3.04i)11-s + (0.257 + 3.43i)13-s + (5.02 − 4.66i)15-s + (−1.92 − 1.31i)17-s + (3.23 + 0.996i)19-s + (−3.68 − 4.61i)21-s + (−4.82 − 4.47i)23-s + (0.529 − 1.34i)25-s + (−3.13 + 1.51i)27-s + (−5.34 − 0.806i)29-s + (−0.778 − 1.98i)31-s + ⋯
L(s)  = 1  + (−1.54 + 0.232i)3-s + (−0.938 + 0.639i)5-s + (0.413 + 0.715i)7-s + (1.36 − 0.421i)9-s + (−0.209 + 0.917i)11-s + (0.0713 + 0.951i)13-s + (1.29 − 1.20i)15-s + (−0.467 − 0.318i)17-s + (0.741 + 0.228i)19-s + (−0.803 − 1.00i)21-s + (−1.00 − 0.933i)23-s + (0.105 − 0.269i)25-s + (−0.603 + 0.290i)27-s + (−0.993 − 0.149i)29-s + (−0.139 − 0.356i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(688\)    =    \(2^{4} \cdot 43\)
Sign: $-0.703 + 0.710i$
Analytic conductor: \(5.49370\)
Root analytic conductor: \(2.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{688} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 688,\ (\ :1/2),\ -0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0539343 - 0.129255i\)
\(L(\frac12)\) \(\approx\) \(0.0539343 - 0.129255i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.03 + 2.55i)T \)
good3 \( 1 + (2.66 - 0.402i)T + (2.86 - 0.884i)T^{2} \)
5 \( 1 + (2.09 - 1.43i)T + (1.82 - 4.65i)T^{2} \)
7 \( 1 + (-1.09 - 1.89i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.694 - 3.04i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.257 - 3.43i)T + (-12.8 + 1.93i)T^{2} \)
17 \( 1 + (1.92 + 1.31i)T + (6.21 + 15.8i)T^{2} \)
19 \( 1 + (-3.23 - 0.996i)T + (15.6 + 10.7i)T^{2} \)
23 \( 1 + (4.82 + 4.47i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (5.34 + 0.806i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (0.778 + 1.98i)T + (-22.7 + 21.0i)T^{2} \)
37 \( 1 + (1.73 - 3.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.33 + 1.66i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (0.260 + 1.14i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-0.777 + 10.3i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (-5.53 + 2.66i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.913 + 2.32i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (9.87 + 3.04i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (8.30 - 7.70i)T + (5.30 - 70.8i)T^{2} \)
73 \( 1 + (-0.624 - 8.33i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + (-4.90 - 8.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.83 + 0.427i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (-14.8 + 2.23i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-0.939 + 4.11i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.23296648695432349844701507453, −10.36057518094168824814618410802, −9.542618975553228423979327565709, −8.317314438396662877841736607065, −7.24411423129636244090520136417, −6.62513758779362485951012792631, −5.56364973607961912705742230750, −4.73308689515081347745221306229, −3.84242647838753819309402373872, −2.07943461660372009211299657696, 0.10321322130043380316505516252, 1.18745355310808643147228748627, 3.53278181521140022369620274660, 4.55309567466421633207071232567, 5.43385837687216147915955921495, 6.14099582952456731704967866024, 7.45086125952658243605268433091, 7.87178721391184339574705850723, 9.013102658814905097243305398762, 10.40086732070854297220721488470

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