Properties

Label 2-7600-1.1-c1-0-110
Degree $2$
Conductor $7600$
Sign $-1$
Analytic cond. $60.6863$
Root an. cond. $7.79014$
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  − 2.41·3-s + 2.41·7-s + 2.82·9-s − 1.41·11-s + 1.82·13-s + 17-s + 19-s − 5.82·21-s − 5.24·23-s + 0.414·27-s + 3.82·29-s − 3.41·31-s + 3.41·33-s + 5.17·37-s − 4.41·39-s − 7.07·41-s − 2.24·43-s − 8·47-s − 1.17·49-s − 2.41·51-s − 1.82·53-s − 2.41·57-s − 14.4·59-s + 3.41·61-s + 6.82·63-s + 6.07·67-s + 12.6·69-s + ⋯
L(s)  = 1  − 1.39·3-s + 0.912·7-s + 0.942·9-s − 0.426·11-s + 0.507·13-s + 0.242·17-s + 0.229·19-s − 1.27·21-s − 1.09·23-s + 0.0797·27-s + 0.710·29-s − 0.613·31-s + 0.594·33-s + 0.850·37-s − 0.706·39-s − 1.10·41-s − 0.341·43-s − 1.16·47-s − 0.167·49-s − 0.338·51-s − 0.251·53-s − 0.319·57-s − 1.87·59-s + 0.437·61-s + 0.860·63-s + 0.741·67-s + 1.52·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7600\)    =    \(2^{4} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(60.6863\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 2.41T + 3T^{2} \)
7 \( 1 - 2.41T + 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 - 1.82T + 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
23 \( 1 + 5.24T + 23T^{2} \)
29 \( 1 - 3.82T + 29T^{2} \)
31 \( 1 + 3.41T + 31T^{2} \)
37 \( 1 - 5.17T + 37T^{2} \)
41 \( 1 + 7.07T + 41T^{2} \)
43 \( 1 + 2.24T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 1.82T + 53T^{2} \)
59 \( 1 + 14.4T + 59T^{2} \)
61 \( 1 - 3.41T + 61T^{2} \)
67 \( 1 - 6.07T + 67T^{2} \)
71 \( 1 + 3.07T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 0.828T + 79T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 + 3.75T + 89T^{2} \)
97 \( 1 - 7.65T + 97T^{2} \)
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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

−7.61771084735817423262311289446, −6.51267623496176993291326644980, −6.22547758500510876648110098454, −5.26710488766648331407377673434, −4.99881897894251650911263695977, −4.16711607889211427801941163521, −3.20673599428109798944918002263, −1.98525114463207343413458412028, −1.13896275890434627643294934888, 0, 1.13896275890434627643294934888, 1.98525114463207343413458412028, 3.20673599428109798944918002263, 4.16711607889211427801941163521, 4.99881897894251650911263695977, 5.26710488766648331407377673434, 6.22547758500510876648110098454, 6.51267623496176993291326644980, 7.61771084735817423262311289446

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