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Properties

Label 2-704-11.9-c1-0-4
Degree $2$
Conductor $704$
Sign $0.353 - 0.935i$
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  + (−0.309 + 0.224i)3-s + (−0.190 − 0.587i)5-s + (−2.30 − 1.67i)7-s + (−0.881 + 2.71i)9-s + (3.23 − 0.726i)11-s + (−1.42 + 4.39i)13-s + (0.190 + 0.138i)15-s + (1.42 + 4.39i)17-s + (−2.30 + 1.67i)19-s + 1.09·21-s + 6.47·23-s + (3.73 − 2.71i)25-s + (−0.690 − 2.12i)27-s + (5.16 + 3.75i)29-s + (−1.80 + 5.56i)31-s + ⋯
L(s)  = 1  + (−0.178 + 0.129i)3-s + (−0.0854 − 0.262i)5-s + (−0.872 − 0.634i)7-s + (−0.293 + 0.904i)9-s + (0.975 − 0.219i)11-s + (−0.395 + 1.21i)13-s + (0.0493 + 0.0358i)15-s + (0.346 + 1.06i)17-s + (−0.529 + 0.384i)19-s + 0.237·21-s + 1.34·23-s + (0.747 − 0.542i)25-s + (−0.132 − 0.409i)27-s + (0.958 + 0.696i)29-s + (−0.324 + 0.999i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $0.353 - 0.935i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ 0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.916116 + 0.632880i\)
\(L(\frac12)\) \(\approx\) \(0.916116 + 0.632880i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 + 0.726i)T \)
good3 \( 1 + (0.309 - 0.224i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.190 + 0.587i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.30 + 1.67i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.42 - 4.39i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.42 - 4.39i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.30 - 1.67i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 + (-5.16 - 3.75i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.80 - 5.56i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.92 - 2.85i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.16 - 3.75i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (2.92 - 2.12i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.19 - 6.74i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.16 + 5.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.42 + 4.39i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.94T + 67T^{2} \)
71 \( 1 + (2.66 + 8.19i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-9.78 - 7.10i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.28 - 13.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.95 - 15.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.47T + 89T^{2} \)
97 \( 1 + (-1.71 + 5.29i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.63585664591381558439776982788, −9.790582669396724766889292823762, −8.899102698986901153826768428968, −8.136637410839600524284847617539, −6.81076851063586642722526889711, −6.44910202063307187353499655421, −5.02672853747684395921581011389, −4.20214456075464380748814655327, −3.08925667266382710451016269368, −1.44102712200036852612189581629, 0.64374974835437508612945788622, 2.72874610953053552109588852412, 3.43343108604074772348455074479, 4.89710558511264514934792430125, 5.95869080046415823698039639943, 6.67403977309142180190183415680, 7.47406410765642805865835880153, 8.836436809480522680509475187530, 9.331242329504481804898027038552, 10.15589161267780004558425739668

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