Properties

Label 2-51376-1.1-c1-0-14
Degree $2$
Conductor $51376$
Sign $-1$
Analytic cond. $410.239$
Root an. cond. $20.2543$
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  + 2·3-s − 3·5-s − 7-s + 9-s + 3·11-s − 6·15-s − 3·17-s + 19-s − 2·21-s + 4·25-s − 4·27-s + 6·29-s − 4·31-s + 6·33-s + 3·35-s − 2·37-s + 6·41-s + 43-s − 3·45-s − 3·47-s − 6·49-s − 6·51-s + 12·53-s − 9·55-s + 2·57-s − 6·59-s − 61-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 1.54·15-s − 0.727·17-s + 0.229·19-s − 0.436·21-s + 4/5·25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s + 1.04·33-s + 0.507·35-s − 0.328·37-s + 0.937·41-s + 0.152·43-s − 0.447·45-s − 0.437·47-s − 6/7·49-s − 0.840·51-s + 1.64·53-s − 1.21·55-s + 0.264·57-s − 0.781·59-s − 0.128·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51376\)    =    \(2^{4} \cdot 13^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(410.239\)
Root analytic conductor: \(20.2543\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51376} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 51376,\ (\ :1/2),\ -1)\)

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

−14.80108882921239, −14.21282845999263, −13.88552734068647, −13.29877898037357, −12.65706960782446, −12.24598662628326, −11.65373281845814, −11.25366320840701, −10.71961406902378, −9.922305147702180, −9.405587638949685, −8.915943705086551, −8.465926374130778, −8.055495989254266, −7.395405282111059, −7.022685728255524, −6.375228062196633, −5.702672220149246, −4.738777453900088, −4.256185073091533, −3.669118724440434, −3.300753583594263, −2.621085062041899, −1.898200738423622, −0.9258474095287729, 0, 0.9258474095287729, 1.898200738423622, 2.621085062041899, 3.300753583594263, 3.669118724440434, 4.256185073091533, 4.738777453900088, 5.702672220149246, 6.375228062196633, 7.022685728255524, 7.395405282111059, 8.055495989254266, 8.465926374130778, 8.915943705086551, 9.405587638949685, 9.922305147702180, 10.71961406902378, 11.25366320840701, 11.65373281845814, 12.24598662628326, 12.65706960782446, 13.29877898037357, 13.88552734068647, 14.21282845999263, 14.80108882921239

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