Properties

Label 2-76e2-1.1-c1-0-26
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·7-s − 2·9-s − 2·11-s − 13-s − 5·17-s − 3·21-s + 23-s − 5·25-s − 5·27-s + 3·29-s + 4·31-s − 2·33-s − 2·37-s − 39-s + 8·41-s + 8·43-s + 8·47-s + 2·49-s − 5·51-s − 9·53-s + 59-s + 14·61-s + 6·63-s + 13·67-s + 69-s + 10·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·7-s − 2/3·9-s − 0.603·11-s − 0.277·13-s − 1.21·17-s − 0.654·21-s + 0.208·23-s − 25-s − 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.348·33-s − 0.328·37-s − 0.160·39-s + 1.24·41-s + 1.21·43-s + 1.16·47-s + 2/7·49-s − 0.700·51-s − 1.23·53-s + 0.130·59-s + 1.79·61-s + 0.755·63-s + 1.58·67-s + 0.120·69-s + 1.18·71-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5776,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.254505077\)
\(L(\frac12)\) \(\approx\) \(1.254505077\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.206640896014532768623911807179, −7.45223598569111326218712477316, −6.68020406568997546924972712188, −6.04451138925848489743249658527, −5.32044629281277041183777124947, −4.31006351639432455417977277039, −3.57486308887784256330172967773, −2.65611179737988448158990271084, −2.28916410595444076068801161106, −0.53700971429406905688036613146, 0.53700971429406905688036613146, 2.28916410595444076068801161106, 2.65611179737988448158990271084, 3.57486308887784256330172967773, 4.31006351639432455417977277039, 5.32044629281277041183777124947, 6.04451138925848489743249658527, 6.68020406568997546924972712188, 7.45223598569111326218712477316, 8.206640896014532768623911807179

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