Properties

Label 2-76e2-1.1-c1-0-145
Degree $2$
Conductor $5776$
Sign $-1$
Analytic cond. $46.1215$
Root an. cond. $6.79128$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s − 2·9-s + 6·11-s − 5·13-s + 3·17-s + 21-s − 3·23-s − 5·25-s − 5·27-s − 9·29-s − 4·31-s + 6·33-s − 2·37-s − 5·39-s − 8·43-s − 6·49-s + 3·51-s + 3·53-s + 9·59-s − 10·61-s − 2·63-s + 5·67-s − 3·69-s − 6·71-s − 7·73-s − 5·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.80·11-s − 1.38·13-s + 0.727·17-s + 0.218·21-s − 0.625·23-s − 25-s − 0.962·27-s − 1.67·29-s − 0.718·31-s + 1.04·33-s − 0.328·37-s − 0.800·39-s − 1.21·43-s − 6/7·49-s + 0.420·51-s + 0.412·53-s + 1.17·59-s − 1.28·61-s − 0.251·63-s + 0.610·67-s − 0.361·69-s − 0.712·71-s − 0.819·73-s − 0.577·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5776\)    =    \(2^{4} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(46.1215\)
Root analytic conductor: \(6.79128\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5776} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5776,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.66995734298328287502507015405, −7.29904094327803276580864474565, −6.29607557013450001946456563759, −5.64317441971502851003917388776, −4.83778114166735308330677292657, −3.83932823582255789112395130179, −3.41239623655734813432880019975, −2.21851462150366027971745653477, −1.59049790592698824897087379644, 0, 1.59049790592698824897087379644, 2.21851462150366027971745653477, 3.41239623655734813432880019975, 3.83932823582255789112395130179, 4.83778114166735308330677292657, 5.64317441971502851003917388776, 6.29607557013450001946456563759, 7.29904094327803276580864474565, 7.66995734298328287502507015405

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