Properties

Label 2-1904-1.1-c1-0-47
Degree $2$
Conductor $1904$
Sign $-1$
Analytic cond. $15.2035$
Root an. cond. $3.89916$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·3-s − 1.30·5-s + 7-s + 2.30·9-s − 4·11-s − 4.60·13-s − 3·15-s − 17-s − 8.60·19-s + 2.30·21-s − 4·23-s − 3.30·25-s − 1.60·27-s + 9.21·29-s − 7.30·31-s − 9.21·33-s − 1.30·35-s + 9.81·37-s − 10.6·39-s − 11.5·41-s + 4.30·43-s − 3.00·45-s + 2.60·47-s + 49-s − 2.30·51-s + 0.697·53-s + 5.21·55-s + ⋯
L(s)  = 1  + 1.32·3-s − 0.582·5-s + 0.377·7-s + 0.767·9-s − 1.20·11-s − 1.27·13-s − 0.774·15-s − 0.242·17-s − 1.97·19-s + 0.502·21-s − 0.834·23-s − 0.660·25-s − 0.308·27-s + 1.71·29-s − 1.31·31-s − 1.60·33-s − 0.220·35-s + 1.61·37-s − 1.69·39-s − 1.79·41-s + 0.656·43-s − 0.447·45-s + 0.380·47-s + 0.142·49-s − 0.322·51-s + 0.0957·53-s + 0.702·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1904\)    =    \(2^{4} \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(15.2035\)
Root analytic conductor: \(3.89916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1904,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
17 \( 1 + T \)
good3 \( 1 - 2.30T + 3T^{2} \)
5 \( 1 + 1.30T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 4.60T + 13T^{2} \)
19 \( 1 + 8.60T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 9.21T + 29T^{2} \)
31 \( 1 + 7.30T + 31T^{2} \)
37 \( 1 - 9.81T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 4.30T + 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 - 0.697T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 15.5T + 61T^{2} \)
67 \( 1 + 2.69T + 67T^{2} \)
71 \( 1 - 3.39T + 71T^{2} \)
73 \( 1 - 7.51T + 73T^{2} \)
79 \( 1 - 2.60T + 79T^{2} \)
83 \( 1 + 3.21T + 83T^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−8.448606049139588454775580671519, −8.226182419599697852830173998777, −7.56038371669997299099939184368, −6.69833741520059123655787074776, −5.45107943684789266421527499456, −4.48339235928316516582873839974, −3.80428142841980332863843142159, −2.53049436939176012175857732273, −2.20543843114629527667720682516, 0, 2.20543843114629527667720682516, 2.53049436939176012175857732273, 3.80428142841980332863843142159, 4.48339235928316516582873839974, 5.45107943684789266421527499456, 6.69833741520059123655787074776, 7.56038371669997299099939184368, 8.226182419599697852830173998777, 8.448606049139588454775580671519

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