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Properties

Label 2-704-11.4-c1-0-4
Degree $2$
Conductor $704$
Sign $0.0694 - 0.997i$
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  + (0.190 − 0.587i)3-s + (−1.30 + 0.951i)5-s + (0.190 + 0.587i)7-s + (2.11 + 1.53i)9-s + (−3.23 + 0.726i)11-s + (0.690 + 0.502i)13-s + (0.309 + 0.951i)15-s + (−3.92 + 2.85i)17-s + (−0.572 + 1.76i)19-s + 0.381·21-s + 4·23-s + (−0.736 + 2.26i)25-s + (2.80 − 2.04i)27-s + (2.57 + 7.91i)29-s + (−8.16 − 5.93i)31-s + ⋯
L(s)  = 1  + (0.110 − 0.339i)3-s + (−0.585 + 0.425i)5-s + (0.0721 + 0.222i)7-s + (0.706 + 0.512i)9-s + (−0.975 + 0.219i)11-s + (0.191 + 0.139i)13-s + (0.0797 + 0.245i)15-s + (−0.952 + 0.691i)17-s + (−0.131 + 0.404i)19-s + 0.0833·21-s + 0.834·23-s + (−0.147 + 0.453i)25-s + (0.540 − 0.392i)27-s + (0.477 + 1.47i)29-s + (−1.46 − 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $0.0694 - 0.997i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ 0.0694 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.814400 + 0.759653i\)
\(L(\frac12)\) \(\approx\) \(0.814400 + 0.759653i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.190 + 0.587i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.30 - 0.951i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.190 - 0.587i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.690 - 0.502i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.92 - 2.85i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.572 - 1.76i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (-2.57 - 7.91i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (8.16 + 5.93i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.28 - 7.02i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.28 - 7.02i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + (2.95 - 9.09i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.30 + 0.951i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.42 + 4.39i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.92 + 2.85i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.52T + 67T^{2} \)
71 \( 1 + (-4.92 + 3.57i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.04 + 9.37i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.30 - 1.67i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.39 - 6.82i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + (2.69 + 1.95i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.90024223665979221901694500288, −9.864088906793680050301182717305, −8.801121890430398515291454889445, −7.88211091381235878820764161946, −7.30647818477623968821930764601, −6.38077005394140132515397574814, −5.14835602141372492087227309525, −4.19380161604062251216377041374, −2.94604717979169455616141608452, −1.72499198700020913123458509048, 0.57860853304774827232541025519, 2.50449470584052816248329512979, 3.84648569751503651028688426898, 4.57826988552891443907818082189, 5.58522336273474374778132283569, 6.91868292996879587219329009569, 7.57025374724597293794274668235, 8.672872027134714002967234055949, 9.232585198370129513869736078720, 10.35634550438651451065639884482

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