Properties

Label 2-1904-1.1-c1-0-39
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  + 0.264·3-s + 0.163·5-s + 7-s − 2.92·9-s − 3.23·11-s + 2.47·13-s + 0.0433·15-s + 17-s − 1.47·19-s + 0.264·21-s − 4.85·23-s − 4.97·25-s − 1.57·27-s + 0.378·29-s − 8.78·31-s − 0.857·33-s + 0.163·35-s + 8.56·37-s + 0.655·39-s + 5.97·41-s − 10.0·43-s − 0.479·45-s + 8.18·47-s + 49-s + 0.264·51-s − 8.72·53-s − 0.529·55-s + ⋯
L(s)  = 1  + 0.152·3-s + 0.0732·5-s + 0.377·7-s − 0.976·9-s − 0.975·11-s + 0.685·13-s + 0.0112·15-s + 0.242·17-s − 0.337·19-s + 0.0578·21-s − 1.01·23-s − 0.994·25-s − 0.302·27-s + 0.0703·29-s − 1.57·31-s − 0.149·33-s + 0.0276·35-s + 1.40·37-s + 0.104·39-s + 0.932·41-s − 1.52·43-s − 0.0715·45-s + 1.19·47-s + 0.142·49-s + 0.0371·51-s − 1.19·53-s − 0.0714·55-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 \( 1 - 0.264T + 3T^{2} \)
5 \( 1 - 0.163T + 5T^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
13 \( 1 - 2.47T + 13T^{2} \)
19 \( 1 + 1.47T + 19T^{2} \)
23 \( 1 + 4.85T + 23T^{2} \)
29 \( 1 - 0.378T + 29T^{2} \)
31 \( 1 + 8.78T + 31T^{2} \)
37 \( 1 - 8.56T + 37T^{2} \)
41 \( 1 - 5.97T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 8.18T + 47T^{2} \)
53 \( 1 + 8.72T + 53T^{2} \)
59 \( 1 + 2.85T + 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 + 4.91T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 5.65T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−8.807929792369688411367772456337, −7.896783376790930471896708280498, −7.62802378590915908978575386777, −6.11416615276011873489365712737, −5.80017717445386214894354939702, −4.75895901353658390582609894907, −3.74180899562977542287645354573, −2.76802031154388122956489060334, −1.76686775996363392713797392868, 0, 1.76686775996363392713797392868, 2.76802031154388122956489060334, 3.74180899562977542287645354573, 4.75895901353658390582609894907, 5.80017717445386214894354939702, 6.11416615276011873489365712737, 7.62802378590915908978575386777, 7.896783376790930471896708280498, 8.807929792369688411367772456337

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