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Properties

Label 2-704-88.69-c1-0-13
Degree $2$
Conductor $704$
Sign $0.966 - 0.256i$
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  + (−0.0422 + 0.0137i)3-s + (−2.42 + 1.76i)9-s + (2.41 − 2.27i)11-s + (5.85 + 4.25i)17-s + (7.08 − 2.30i)19-s + (1.54 + 4.75i)25-s + (0.156 − 0.215i)27-s + (−0.0709 + 0.129i)33-s + (−0.554 − 1.70i)41-s − 3.10i·43-s + (5.66 + 4.11i)49-s + (−0.305 − 0.0994i)51-s + (−0.268 + 0.194i)57-s + (11.0 + 3.58i)59-s − 12.3i·67-s + ⋯
L(s)  = 1  + (−0.0244 + 0.00793i)3-s + (−0.808 + 0.587i)9-s + (0.728 − 0.685i)11-s + (1.41 + 1.03i)17-s + (1.62 − 0.528i)19-s + (0.309 + 0.951i)25-s + (0.0301 − 0.0415i)27-s + (−0.0123 + 0.0225i)33-s + (−0.0866 − 0.266i)41-s − 0.473i·43-s + (0.809 + 0.587i)49-s + (−0.0428 − 0.0139i)51-s + (−0.0355 + 0.0258i)57-s + (1.43 + 0.466i)59-s − 1.51i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53137 + 0.199940i\)
\(L(\frac12)\) \(\approx\) \(1.53137 + 0.199940i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 + 2.27i)T \)
good3 \( 1 + (0.0422 - 0.0137i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.85 - 4.25i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-7.08 + 2.30i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.554 + 1.70i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.10iT - 43T^{2} \)
47 \( 1 + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-11.0 - 3.58i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.58 - 11.0i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.58 - 10.4i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 17.8T + 89T^{2} \)
97 \( 1 + (-14.6 + 10.6i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.53965295073930556574092335379, −9.578716921537746176576695605005, −8.738653031019065569583830669664, −7.932287714340753548950306933290, −7.04108034025827930795300611506, −5.78106977875971603205113572676, −5.31236912388981103214634651445, −3.79183232632950197918740629676, −2.91577562184134370011547684227, −1.23208237037704920447254197833, 1.05563560105422264859844092152, 2.79917119792363199882366083526, 3.74922867779951967880819323400, 5.06405791857480427934032959374, 5.88161341562849275436997071806, 6.94536336874312972097060566188, 7.73925618741305399829891519636, 8.775199619630545326519784616006, 9.639386696080172543496969403218, 10.15205146818534316135799077607

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