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Properties

Label 2-704-88.37-c1-0-22
Degree $2$
Conductor $704$
Sign $-0.656 - 0.754i$
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  + (−2.48 − 0.809i)3-s + (−1.80 − 2.47i)5-s + (−1.53 − 4.71i)7-s + (3.11 + 2.26i)9-s + (−0.726 − 3.23i)11-s + (1.80 − 2.47i)13-s + (2.47 + 7.62i)15-s + (0.309 − 0.224i)17-s + (−4.39 − 1.42i)19-s + 12.9i·21-s + 6.12·23-s + (−1.35 + 4.16i)25-s + (−1.31 − 1.80i)27-s + (−1.11 + 0.361i)29-s + (−5.54 − 4.02i)31-s + ⋯
L(s)  = 1  + (−1.43 − 0.467i)3-s + (−0.805 − 1.10i)5-s + (−0.578 − 1.78i)7-s + (1.03 + 0.755i)9-s + (−0.219 − 0.975i)11-s + (0.499 − 0.687i)13-s + (0.639 + 1.96i)15-s + (0.0749 − 0.0544i)17-s + (−1.00 − 0.327i)19-s + 2.83i·21-s + 1.27·23-s + (−0.270 + 0.833i)25-s + (−0.252 − 0.348i)27-s + (−0.206 + 0.0671i)29-s + (−0.995 − 0.723i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-0.656 - 0.754i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ -0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.203408 + 0.446503i\)
\(L(\frac12)\) \(\approx\) \(0.203408 + 0.446503i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.726 + 3.23i)T \)
good3 \( 1 + (2.48 + 0.809i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.80 + 2.47i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.53 + 4.71i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.80 + 2.47i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.309 + 0.224i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.39 + 1.42i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.12T + 23T^{2} \)
29 \( 1 + (1.11 - 0.361i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.54 + 4.02i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-4.71 + 1.53i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.954 + 2.93i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.76iT - 43T^{2} \)
47 \( 1 + (0.361 - 1.11i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.02 + 5.54i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-7.46 + 2.42i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.02 - 5.54i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 9.70iT - 67T^{2} \)
71 \( 1 + (-10.4 + 7.62i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.66 + 11.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.584 - 0.425i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.85 - 3.92i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.52T + 89T^{2} \)
97 \( 1 + (-2.54 - 1.84i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.24339613470850166351591696668, −8.952320194774170673429647674066, −7.958600722348231366127819973841, −7.19031164407993205827741057821, −6.33111374945909429491976592414, −5.38005918815260708878808228863, −4.43159860300778060620602049799, −3.55095021827633118190160173552, −0.900441087789663418991506429486, −0.43357175315419671781720291187, 2.33273891035311400174436891634, 3.60545336301230398007291421593, 4.77549375047253610767031874856, 5.70135476120122576034789548103, 6.49703408560385210140328948157, 7.11636064996529648651940167787, 8.524251056423610589076402053161, 9.438895952833776968142375212058, 10.34909612541964442191076787110, 11.14611980884478723328653663014

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