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Properties

Label 2-704-11.10-c0-0-0
Degree $2$
Conductor $704$
Sign $1$
Analytic cond. $0.351341$
Root an. cond. $0.592740$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 11-s − 15-s + 23-s + 27-s + 31-s − 33-s + 37-s − 2·47-s + 49-s − 2·53-s + 55-s − 59-s − 67-s − 69-s + 71-s − 81-s − 89-s − 93-s − 97-s − 2·103-s − 111-s − 113-s + 115-s + ⋯
L(s)  = 1  − 3-s + 5-s + 11-s − 15-s + 23-s + 27-s + 31-s − 33-s + 37-s − 2·47-s + 49-s − 2·53-s + 55-s − 59-s − 67-s − 69-s + 71-s − 81-s − 89-s − 93-s − 97-s − 2·103-s − 111-s − 113-s + 115-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.351341\)
Root analytic conductor: \(0.592740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{704} (65, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8342109324\)
\(L(\frac12)\) \(\approx\) \(0.8342109324\)
\(L(1)\) 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 \( 1 - T \)
good3 \( 1 + T + T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 - T + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 - T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 + T )^{2} \)
53 \( ( 1 + T )^{2} \)
59 \( 1 + T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
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

−10.76986502874820360446020620007, −9.775517195333216833439651901402, −9.189706642100871132128751495689, −8.087796887651203646924886576649, −6.71158914396420532070739687727, −6.24407165765924451166630290208, −5.38319360534408517236765833403, −4.46317942736026664517383129516, −2.93434243358027300820327152618, −1.39587762472223235785737155063, 1.39587762472223235785737155063, 2.93434243358027300820327152618, 4.46317942736026664517383129516, 5.38319360534408517236765833403, 6.24407165765924451166630290208, 6.71158914396420532070739687727, 8.087796887651203646924886576649, 9.189706642100871132128751495689, 9.775517195333216833439651901402, 10.76986502874820360446020620007

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