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Properties

Label 2-704-11.3-c1-0-11
Degree $2$
Conductor $704$
Sign $0.927 - 0.374i$
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  + (0.118 + 0.363i)3-s + (2.61 + 1.90i)5-s + (0.618 − 1.90i)7-s + (2.30 − 1.67i)9-s + (0.309 + 3.30i)11-s + (1 − 0.726i)13-s + (−0.381 + 1.17i)15-s + (0.5 + 0.363i)17-s + (−1.80 − 5.56i)19-s + 0.763·21-s − 1.23·23-s + (1.69 + 5.20i)25-s + (1.80 + 1.31i)27-s + (−1.38 + 4.25i)29-s + (1.61 − 1.17i)31-s + ⋯
L(s)  = 1  + (0.0681 + 0.209i)3-s + (1.17 + 0.850i)5-s + (0.233 − 0.718i)7-s + (0.769 − 0.559i)9-s + (0.0931 + 0.995i)11-s + (0.277 − 0.201i)13-s + (−0.0986 + 0.303i)15-s + (0.121 + 0.0881i)17-s + (−0.415 − 1.27i)19-s + 0.166·21-s − 0.257·23-s + (0.338 + 1.04i)25-s + (0.348 + 0.252i)27-s + (−0.256 + 0.789i)29-s + (0.290 − 0.211i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01273 + 0.390943i\)
\(L(\frac12)\) \(\approx\) \(2.01273 + 0.390943i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 3.30i)T \)
good3 \( 1 + (-0.118 - 0.363i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.61 - 1.90i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.618 + 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1 + 0.726i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.5 - 0.363i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.23T + 23T^{2} \)
29 \( 1 + (1.38 - 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.61 + 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.14 + 3.52i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.73 - 5.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.56T + 43T^{2} \)
47 \( 1 + (-2 - 6.15i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.23 - 0.898i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.66 - 8.19i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2 + 1.45i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + (4.23 + 3.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.20 + 9.87i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.8 + 7.88i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.54 + 5.48i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.09T + 89T^{2} \)
97 \( 1 + (5.78 - 4.20i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.45796263000477255791782624319, −9.709126456513467266355842838340, −9.119812211891116903703166691643, −7.66782436890770610737793173452, −6.86593511442014058207956331485, −6.28581107934889659182686803034, −4.96375545083315788195811752620, −4.02059519121880717507133596323, −2.73149174884401650336538326864, −1.48894032084849305696760035562, 1.39336544754811113656699097141, 2.28737442491466018081406974505, 3.92845744287755153731773295868, 5.19422661765513422226681008222, 5.77377233876305941554149123651, 6.68332659587717983714232691594, 8.145976137612683929787730298673, 8.531683897154999850758051211526, 9.595217841023178047537425727696, 10.17087732892897904865455711655

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