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Properties

Label 2-704-44.43-c1-0-7
Degree $2$
Conductor $704$
Sign $-i$
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  + 2.52i·3-s + 4.37·5-s − 3.37·9-s + 3.31i·11-s + 11.0i·15-s − 9.45i·23-s + 14.1·25-s − 0.939i·27-s + 0.644i·31-s − 8.37·33-s − 5.11·37-s − 14.7·45-s + 6.63i·47-s − 7·49-s − 6·53-s + ⋯
L(s)  = 1  + 1.45i·3-s + 1.95·5-s − 1.12·9-s + 1.00i·11-s + 2.84i·15-s − 1.97i·23-s + 2.82·25-s − 0.180i·27-s + 0.115i·31-s − 1.45·33-s − 0.841·37-s − 2.19·45-s + 0.967i·47-s − 49-s − 0.824·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-i$
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: \((2,\ 704,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46875 + 1.46875i\)
\(L(\frac12)\) \(\approx\) \(1.46875 + 1.46875i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.31iT \)
good3 \( 1 - 2.52iT - 3T^{2} \)
5 \( 1 - 4.37T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 9.45iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 0.644iT - 31T^{2} \)
37 \( 1 + 5.11T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 6.28iT - 67T^{2} \)
71 \( 1 + 5.69iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 9.86T + 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−10.37601259009001753734021325791, −9.838045603589953454015357939436, −9.306899815429104416644807952960, −8.488354608314140855573554866012, −6.86274583783224106016723721337, −6.05420586410217754925728117394, −5.03717445339474407680196915613, −4.52121987168902067654342974922, −3.00033106646309711977533788855, −1.89650285076714457536701544820, 1.23755925305209312196101703279, 2.04251960806278962926392859041, 3.18481079677874530557982450803, 5.26783565647143304400367488667, 5.90617625371826688639988007494, 6.57199379865690091315415015121, 7.45279336924087024967902506174, 8.516670992555040166015624096093, 9.317625777635347534316216709933, 10.14452441148956400340341572576

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