Properties

Label 2-88e2-1.1-c1-0-195
Degree $2$
Conductor $7744$
Sign $-1$
Analytic cond. $61.8361$
Root an. cond. $7.86359$
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 + 3·5-s + 2·7-s − 2·9-s − 4·13-s + 3·15-s − 6·17-s − 8·19-s + 2·21-s + 3·23-s + 4·25-s − 5·27-s − 5·31-s + 6·35-s + 37-s − 4·39-s + 10·43-s − 6·45-s − 3·49-s − 6·51-s + 6·53-s − 8·57-s + 3·59-s − 4·61-s − 4·63-s − 12·65-s − 67-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 0.755·7-s − 2/3·9-s − 1.10·13-s + 0.774·15-s − 1.45·17-s − 1.83·19-s + 0.436·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s − 0.898·31-s + 1.01·35-s + 0.164·37-s − 0.640·39-s + 1.52·43-s − 0.894·45-s − 3/7·49-s − 0.840·51-s + 0.824·53-s − 1.05·57-s + 0.390·59-s − 0.512·61-s − 0.503·63-s − 1.48·65-s − 0.122·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7744\)    =    \(2^{6} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(61.8361\)
Root analytic conductor: \(7.86359\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7744,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−7.54166862432157620857053020596, −6.79854618519138189258920039988, −6.09230827167000814448233726197, −5.45156827262255731355713819112, −4.70542687537026569204273011302, −4.04116339581049737147096754950, −2.66574560392864207944710081183, −2.33981349809658735343379181335, −1.66047769005307141108341823826, 0, 1.66047769005307141108341823826, 2.33981349809658735343379181335, 2.66574560392864207944710081183, 4.04116339581049737147096754950, 4.70542687537026569204273011302, 5.45156827262255731355713819112, 6.09230827167000814448233726197, 6.79854618519138189258920039988, 7.54166862432157620857053020596

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