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Properties

Label 2-704-44.43-c1-0-14
Degree $2$
Conductor $704$
Sign $0.979 - 0.200i$
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  + 0.529i·3-s + 2.77·5-s + 3.18·7-s + 2.71·9-s + (−0.665 − 3.24i)11-s + 6.00i·13-s + 1.47i·15-s − 4.67i·17-s − 7.50·19-s + 1.68i·21-s − 5.02i·23-s + 2.71·25-s + 3.02i·27-s − 5.03i·29-s + 3.96i·31-s + ⋯
L(s)  = 1  + 0.305i·3-s + 1.24·5-s + 1.20·7-s + 0.906·9-s + (−0.200 − 0.979i)11-s + 1.66i·13-s + 0.379i·15-s − 1.13i·17-s − 1.72·19-s + 0.367i·21-s − 1.04i·23-s + 0.543·25-s + 0.582i·27-s − 0.934i·29-s + 0.712i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $0.979 - 0.200i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ 0.979 - 0.200i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14137 + 0.217095i\)
\(L(\frac12)\) \(\approx\) \(2.14137 + 0.217095i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.665 + 3.24i)T \)
good3 \( 1 - 0.529iT - 3T^{2} \)
5 \( 1 - 2.77T + 5T^{2} \)
7 \( 1 - 3.18T + 7T^{2} \)
13 \( 1 - 6.00iT - 13T^{2} \)
17 \( 1 + 4.67iT - 17T^{2} \)
19 \( 1 + 7.50T + 19T^{2} \)
23 \( 1 + 5.02iT - 23T^{2} \)
29 \( 1 + 5.03iT - 29T^{2} \)
31 \( 1 - 3.96iT - 31T^{2} \)
37 \( 1 - 2.77T + 37T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 + 1.33T + 43T^{2} \)
47 \( 1 - 8.61iT - 47T^{2} \)
53 \( 1 + 0.117T + 53T^{2} \)
59 \( 1 - 6.41iT - 59T^{2} \)
61 \( 1 - 4.32iT - 61T^{2} \)
67 \( 1 + 10.5iT - 67T^{2} \)
71 \( 1 + 1.91iT - 71T^{2} \)
73 \( 1 + 7.34iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 7.69T + 83T^{2} \)
89 \( 1 - 1.95T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−10.51612945438549081938233056597, −9.549307077459581680617486023263, −8.922843749103120098440567144340, −7.996576914137566652498369622847, −6.75475021677604700445026989890, −6.08834944452247273417042770294, −4.80655844333501687525680107253, −4.32731984837008455319759556108, −2.47486754395673199657401570648, −1.52478090505571512276035705062, 1.55625388316012775600523953122, 2.20467390999478138886652723049, 4.02032363021776380409839334273, 5.11500538561328528462017081882, 5.83942654224657105983761738718, 6.93137181586759514391867270812, 7.85212039702117758248504520683, 8.553357167444585124891030291108, 9.811453765148303671411229099870, 10.33796763678360135324744529473

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