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Properties

Label 2-704-88.21-c0-0-4
Degree $2$
Conductor $704$
Sign $-0.258 + 0.965i$
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  − i·3-s − 1.73i·5-s + i·11-s − 1.73·15-s − 1.73·23-s − 1.99·25-s − i·27-s + 1.73·31-s + 33-s + 1.73i·37-s + 49-s + 1.73·55-s − i·59-s + i·67-s + 1.73i·69-s + ⋯
L(s)  = 1  − i·3-s − 1.73i·5-s + i·11-s − 1.73·15-s − 1.73·23-s − 1.99·25-s − i·27-s + 1.73·31-s + 33-s + 1.73i·37-s + 49-s + 1.73·55-s − i·59-s + i·67-s + 1.73i·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(0.351341\)
Root analytic conductor: \(0.592740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :0),\ -0.258 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9486225243\)
\(L(\frac12)\) \(\approx\) \(0.9486225243\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 + iT - T^{2} \)
5 \( 1 + 1.73iT - T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.73T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.73T + T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.08040698691470360840492460271, −9.604008416826061200504921776161, −8.340571488010252642225880465354, −8.043359828562033382978911581842, −6.93407456699106966502233083049, −5.98962888256222667869782967086, −4.85720412065493504939454193658, −4.16609623914396943708179056102, −2.17212466483325350060575125605, −1.16728486179106102319791020140, 2.44568619602551735819792948858, 3.48722383147019187723139050942, 4.19158621722119490265550981881, 5.65899472335893004724788390613, 6.40521337497273557715870681164, 7.37671498738757632703418864884, 8.314266631517820958403245538905, 9.492945756502734424269664877455, 10.20745764341117358145841610966, 10.72484547208088205797295849686

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