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Properties

Label 2-704-88.83-c1-0-5
Degree $2$
Conductor $704$
Sign $-0.674 - 0.738i$
Analytic cond. $5.62146$
Root an. cond. $2.37096$
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  + (2.11 + 1.53i)3-s + (−1.17 − 0.381i)5-s + (−3.07 + 2.23i)7-s + (1.19 + 3.66i)9-s + (−2.80 + 1.76i)11-s + (−1.90 − 2.61i)15-s + (3.35 + 1.08i)17-s + (−2.07 + 2.85i)19-s − 9.95·21-s + 6i·23-s + (−2.80 − 2.04i)25-s + (−0.690 + 2.12i)27-s + (−4.25 + 3.09i)29-s + (6.88 − 2.23i)31-s + (−8.66 − 0.587i)33-s + ⋯
L(s)  = 1  + (1.22 + 0.888i)3-s + (−0.525 − 0.170i)5-s + (−1.16 + 0.845i)7-s + (0.396 + 1.22i)9-s + (−0.846 + 0.531i)11-s + (−0.491 − 0.675i)15-s + (0.813 + 0.264i)17-s + (−0.475 + 0.654i)19-s − 2.17·21-s + 1.25i·23-s + (−0.561 − 0.408i)25-s + (−0.132 + 0.409i)27-s + (−0.789 + 0.573i)29-s + (1.23 − 0.401i)31-s + (−1.50 − 0.102i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-0.674 - 0.738i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ -0.674 - 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574477 + 1.30184i\)
\(L(\frac12)\) \(\approx\) \(0.574477 + 1.30184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (2.80 - 1.76i)T \)
good3 \( 1 + (-2.11 - 1.53i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (1.17 + 0.381i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (3.07 - 2.23i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.35 - 1.08i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.07 - 2.85i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + (4.25 - 3.09i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-6.88 + 2.23i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.17 - 3i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.35 + 4.61i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 12.7iT - 43T^{2} \)
47 \( 1 + (-5.70 + 7.85i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.07 - i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.39 + 6.10i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.52 + 10.8i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 + (-11.4 - 3.70i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.590 + 0.812i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.45 - 4.47i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.427 + 0.138i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 1.85T + 89T^{2} \)
97 \( 1 + (-2.57 - 7.91i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.32045409149158447937619964151, −9.767262066533916556223228807950, −9.201355838601455622256566792377, −8.178059126667805983489726358365, −7.70838320909545344309751359477, −6.25717881535089602159353792631, −5.21269034509867009846261279524, −3.99317153873888929063525718285, −3.27503150296644360062005141052, −2.32173378986605508682491081513, 0.61783188027387094603572973085, 2.51854160385647139069969964040, 3.23004168097470182971668194535, 4.21612983338860037628170059177, 5.90052801333211004223245654161, 6.95581919951927654898409282197, 7.50781584046559185987019879526, 8.250806473399698859689067360079, 9.102709125953566633052105452565, 10.07347092403812978773815201524

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