Properties

Label 2-1904-1.1-c1-0-37
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  − 2.48·3-s + 4.02·5-s + 7-s + 3.18·9-s + 1.23·11-s − 6.47·13-s − 10.0·15-s + 17-s − 6.97·19-s − 2.48·21-s − 7.07·23-s + 11.1·25-s − 0.462·27-s − 6.31·29-s − 4.88·31-s − 3.07·33-s + 4.02·35-s − 2.63·37-s + 16.0·39-s − 10.1·41-s + 2.69·43-s + 12.8·45-s + 3.67·47-s + 49-s − 2.48·51-s − 1.39·53-s + 4.97·55-s + ⋯
L(s)  = 1  − 1.43·3-s + 1.79·5-s + 0.377·7-s + 1.06·9-s + 0.372·11-s − 1.79·13-s − 2.58·15-s + 0.242·17-s − 1.60·19-s − 0.542·21-s − 1.47·23-s + 2.23·25-s − 0.0890·27-s − 1.17·29-s − 0.877·31-s − 0.535·33-s + 0.680·35-s − 0.433·37-s + 2.57·39-s − 1.59·41-s + 0.411·43-s + 1.91·45-s + 0.536·47-s + 0.142·49-s − 0.348·51-s − 0.190·53-s + 0.670·55-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 \( 1 + 2.48T + 3T^{2} \)
5 \( 1 - 4.02T + 5T^{2} \)
11 \( 1 - 1.23T + 11T^{2} \)
13 \( 1 + 6.47T + 13T^{2} \)
19 \( 1 + 6.97T + 19T^{2} \)
23 \( 1 + 7.07T + 23T^{2} \)
29 \( 1 + 6.31T + 29T^{2} \)
31 \( 1 + 4.88T + 31T^{2} \)
37 \( 1 + 2.63T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 2.69T + 43T^{2} \)
47 \( 1 - 3.67T + 47T^{2} \)
53 \( 1 + 1.39T + 53T^{2} \)
59 \( 1 + 5.07T + 59T^{2} \)
61 \( 1 - 3.93T + 61T^{2} \)
67 \( 1 - 5.63T + 67T^{2} \)
71 \( 1 - 6.84T + 71T^{2} \)
73 \( 1 - 0.246T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 - 6.65T + 83T^{2} \)
89 \( 1 - 2.27T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−9.153335495064340454508953032039, −7.950551000765592870196713801796, −6.80242226107146098264162871806, −6.39995404412102598077196338387, −5.40075801317743351795784418323, −5.24990581075479334580043275570, −4.13594955239238313071468468496, −2.33642005094903179380584020249, −1.70812417877573680181152914734, 0, 1.70812417877573680181152914734, 2.33642005094903179380584020249, 4.13594955239238313071468468496, 5.24990581075479334580043275570, 5.40075801317743351795784418323, 6.39995404412102598077196338387, 6.80242226107146098264162871806, 7.950551000765592870196713801796, 9.153335495064340454508953032039

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