Properties

Label 2-1904-1.1-c1-0-22
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  − 2.88·3-s − 1.78·5-s + 7-s + 5.31·9-s − 3.23·11-s + 2.47·13-s + 5.13·15-s + 17-s − 7.76·19-s − 2.88·21-s + 5.32·23-s − 1.82·25-s − 6.66·27-s + 10.5·29-s + 9.64·31-s + 9.32·33-s − 1.78·35-s − 1.62·37-s − 7.12·39-s + 2.82·41-s − 0.587·43-s − 9.46·45-s − 12.1·47-s + 49-s − 2.88·51-s + 3.40·53-s + 5.76·55-s + ⋯
L(s)  = 1  − 1.66·3-s − 0.796·5-s + 0.377·7-s + 1.77·9-s − 0.975·11-s + 0.685·13-s + 1.32·15-s + 0.242·17-s − 1.78·19-s − 0.629·21-s + 1.11·23-s − 0.365·25-s − 1.28·27-s + 1.96·29-s + 1.73·31-s + 1.62·33-s − 0.301·35-s − 0.266·37-s − 1.14·39-s + 0.441·41-s − 0.0895·43-s − 1.41·45-s − 1.77·47-s + 0.142·49-s − 0.403·51-s + 0.467·53-s + 0.777·55-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 \( 1 + 2.88T + 3T^{2} \)
5 \( 1 + 1.78T + 5T^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
13 \( 1 - 2.47T + 13T^{2} \)
19 \( 1 + 7.76T + 19T^{2} \)
23 \( 1 - 5.32T + 23T^{2} \)
29 \( 1 - 10.5T + 29T^{2} \)
31 \( 1 - 9.64T + 31T^{2} \)
37 \( 1 + 1.62T + 37T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 + 0.587T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 3.40T + 53T^{2} \)
59 \( 1 - 7.32T + 59T^{2} \)
61 \( 1 + 3.82T + 61T^{2} \)
67 \( 1 + 0.394T + 67T^{2} \)
71 \( 1 - 2.15T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 + 4.18T + 79T^{2} \)
83 \( 1 + 8.42T + 83T^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 + 7.72T + 97T^{2} \)
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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

−8.486509452241493828604614211614, −8.151193714722178638199191758687, −6.98969649861859894386352973388, −6.42483563759083864951338429728, −5.59807817860182320034012050669, −4.70565326152272854436276361643, −4.26364930555437060036039625304, −2.81570386255833249466026289327, −1.18682378364365594590547214346, 0, 1.18682378364365594590547214346, 2.81570386255833249466026289327, 4.26364930555437060036039625304, 4.70565326152272854436276361643, 5.59807817860182320034012050669, 6.42483563759083864951338429728, 6.98969649861859894386352973388, 8.151193714722178638199191758687, 8.486509452241493828604614211614

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