Properties

Degree $2$
Conductor $49$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s − 3·9-s + 4·11-s − 16-s − 3·18-s + 4·22-s + 8·23-s − 5·25-s + 2·29-s + 5·32-s + 3·36-s − 6·37-s − 12·43-s − 4·44-s + 8·46-s − 5·50-s − 10·53-s + 2·58-s + 7·64-s + 4·67-s + 16·71-s + 9·72-s − 6·74-s + 8·79-s + 9·81-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 9-s + 1.20·11-s − 1/4·16-s − 0.707·18-s + 0.852·22-s + 1.66·23-s − 25-s + 0.371·29-s + 0.883·32-s + 1/2·36-s − 0.986·37-s − 1.82·43-s − 0.603·44-s + 1.17·46-s − 0.707·50-s − 1.37·53-s + 0.262·58-s + 7/8·64-s + 0.488·67-s + 1.89·71-s + 1.06·72-s − 0.697·74-s + 0.900·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{49} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966655\)
\(L(\frac12)\) \(\approx\) \(0.966655\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−15.25649719028727745654998030256, −14.36789003254179224985767994212, −13.55829126515041538892762617601, −12.27943344760252552732176729088, −11.30850265031875128052013945557, −9.489525085039792097687572288594, −8.498120181782134288657898916471, −6.47803659589426412986704519993, −5.08673463814783736975452386140, −3.45773984941604093394269964405, 3.45773984941604093394269964405, 5.08673463814783736975452386140, 6.47803659589426412986704519993, 8.498120181782134288657898916471, 9.489525085039792097687572288594, 11.30850265031875128052013945557, 12.27943344760252552732176729088, 13.55829126515041538892762617601, 14.36789003254179224985767994212, 15.25649719028727745654998030256

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