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Properties

Label 2-704-88.85-c0-0-1
Degree $2$
Conductor $704$
Sign $0.920 + 0.390i$
Analytic cond. $0.351341$
Root an. cond. $0.592740$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.53 − 0.5i)3-s + (1.30 − 0.951i)9-s + (−0.951 + 0.309i)11-s + (−1.11 + 1.53i)17-s + (−0.587 − 1.80i)19-s + (0.309 + 0.951i)25-s + (0.587 − 0.809i)27-s + (−1.30 + 0.951i)33-s + (1.11 − 0.363i)41-s − 1.17·43-s + (−0.809 − 0.587i)49-s + (−0.951 + 2.92i)51-s + (−1.80 − 2.48i)57-s + (0.587 + 0.190i)59-s − 0.618i·67-s + ⋯
L(s)  = 1  + (1.53 − 0.5i)3-s + (1.30 − 0.951i)9-s + (−0.951 + 0.309i)11-s + (−1.11 + 1.53i)17-s + (−0.587 − 1.80i)19-s + (0.309 + 0.951i)25-s + (0.587 − 0.809i)27-s + (−1.30 + 0.951i)33-s + (1.11 − 0.363i)41-s − 1.17·43-s + (−0.809 − 0.587i)49-s + (−0.951 + 2.92i)51-s + (−1.80 − 2.48i)57-s + (0.587 + 0.190i)59-s − 0.618i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $0.920 + 0.390i$
Analytic conductor: \(0.351341\)
Root analytic conductor: \(0.592740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :0),\ 0.920 + 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.417445999\)
\(L(\frac12)\) \(\approx\) \(1.417445999\)
\(L(1)\) 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 \( 1 + (0.951 - 0.309i)T \)
good3 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.17T + T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + 0.618iT - T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
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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

−10.54539280347097069003229309323, −9.457181180248231593311753352992, −8.738466348109959410565296833483, −8.157819261450491197820265845406, −7.24248955941111477660364577425, −6.50067977235065459003635266848, −4.98947340631586030956821227230, −3.88027994101640328194529853726, −2.73854217878456040982319849484, −1.93063747243359873945037668311, 2.19254121162169849704352626684, 3.00883912674624056618382989481, 4.08504559346995318994315126299, 5.01360026659336682333377418600, 6.37382814567628078492946437223, 7.59551927074327854895305245086, 8.210260522272242799907450978623, 8.918003972252193123678902972035, 9.786716346904762190624925203024, 10.40081387136404411683837075625

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