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Properties

Label 2-704-88.21-c0-0-0
Degree $2$
Conductor $704$
Sign $-0.707 - 0.707i$
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  + 2i·3-s − 3·9-s + i·11-s + 25-s − 4i·27-s − 2·33-s + 49-s + 2i·59-s − 2i·67-s + 2i·75-s + 5·81-s + 2·89-s + 2·97-s − 3i·99-s − 2·113-s + ⋯
L(s)  = 1  + 2i·3-s − 3·9-s + i·11-s + 25-s − 4i·27-s − 2·33-s + 49-s + 2i·59-s − 2i·67-s + 2i·75-s + 5·81-s + 2·89-s + 2·97-s − 3i·99-s − 2·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-0.707 - 0.707i$
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: \((2,\ 704,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8560095350\)
\(L(\frac12)\) \(\approx\) \(0.8560095350\)
\(L(1)\) 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 \( 1 - 2iT - T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−10.60478968470828465751359691469, −10.24453960502461917383461263218, −9.292326388280502841526739036008, −8.827601541338360023306098495008, −7.67293772839932259760519253474, −6.31962512994760553433904220611, −5.23468559375425680051641178650, −4.58412574685565488541534096183, −3.70641602444428921028491201471, −2.60022428458752788006475327747, 0.981826750872051918702521062903, 2.32624266104856521471996749791, 3.34745578727143355111942207536, 5.24926264461355859652897330943, 6.12647541872302220434777393475, 6.81604010288850754673375959419, 7.66774653513522675185491628826, 8.426438275231046570272118518283, 9.066422904354898716585572466241, 10.62580710666441373309431785252

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