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Properties

Label 2-704-44.43-c3-0-32
Degree $2$
Conductor $704$
Sign $0.843 - 0.537i$
Analytic cond. $41.5373$
Root an. cond. $6.44494$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.51i·3-s − 7.48·5-s + 35.3·7-s + 24.6·9-s + (19.5 + 30.7i)11-s + 35.0i·13-s + 11.3i·15-s − 61.0i·17-s + 21.7·19-s − 53.6i·21-s + 185. i·23-s − 68.9·25-s − 78.4i·27-s + 214. i·29-s − 140. i·31-s + ⋯
L(s)  = 1  − 0.292i·3-s − 0.669·5-s + 1.90·7-s + 0.914·9-s + (0.537 + 0.843i)11-s + 0.748i·13-s + 0.195i·15-s − 0.870i·17-s + 0.262·19-s − 0.557i·21-s + 1.68i·23-s − 0.551·25-s − 0.559i·27-s + 1.37i·29-s − 0.814i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $0.843 - 0.537i$
Analytic conductor: \(41.5373\)
Root analytic conductor: \(6.44494\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :3/2),\ 0.843 - 0.537i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.469241114\)
\(L(\frac12)\) \(\approx\) \(2.469241114\)
\(L(\frac{5}{2})\) 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 \( 1 + (-19.5 - 30.7i)T \)
good3 \( 1 + 1.51iT - 27T^{2} \)
5 \( 1 + 7.48T + 125T^{2} \)
7 \( 1 - 35.3T + 343T^{2} \)
13 \( 1 - 35.0iT - 2.19e3T^{2} \)
17 \( 1 + 61.0iT - 4.91e3T^{2} \)
19 \( 1 - 21.7T + 6.85e3T^{2} \)
23 \( 1 - 185. iT - 1.21e4T^{2} \)
29 \( 1 - 214. iT - 2.43e4T^{2} \)
31 \( 1 + 140. iT - 2.97e4T^{2} \)
37 \( 1 + 138.T + 5.06e4T^{2} \)
41 \( 1 + 35.0iT - 6.89e4T^{2} \)
43 \( 1 + 367.T + 7.95e4T^{2} \)
47 \( 1 - 123. iT - 1.03e5T^{2} \)
53 \( 1 - 607.T + 1.48e5T^{2} \)
59 \( 1 + 324. iT - 2.05e5T^{2} \)
61 \( 1 - 751. iT - 2.26e5T^{2} \)
67 \( 1 + 642. iT - 3.00e5T^{2} \)
71 \( 1 - 1.03e3iT - 3.57e5T^{2} \)
73 \( 1 - 687. iT - 3.89e5T^{2} \)
79 \( 1 + 65.3T + 4.93e5T^{2} \)
83 \( 1 - 222.T + 5.71e5T^{2} \)
89 \( 1 - 1.31e3T + 7.04e5T^{2} \)
97 \( 1 + 626.T + 9.12e5T^{2} \)
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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

−10.09886279393128688603826668331, −9.218863779004937553361543099420, −8.241875038906424427860767931862, −7.29658439081467700161799819747, −7.13978894963653396018580697753, −5.38643868829354118035782738224, −4.58426746462550500590921452591, −3.83668667376882000369491644928, −1.95384422667553414795177715367, −1.27613642768339902553569288455, 0.793180524936897090882177120458, 1.95481129369673229054891140216, 3.63674871029633442852318285773, 4.41287040393025838348087452853, 5.22521801107725444863277511520, 6.43063116288868611714030438872, 7.64330303899661981001884923931, 8.196672421233831430348141920651, 8.837889001979958720924489079644, 10.30363998117765249499648871909

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