Properties

Label 2-7-7.6-c6-0-1
Degree $2$
Conductor $7$
Sign $1$
Analytic cond. $1.61037$
Root an. cond. $1.26900$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·2-s + 17·4-s − 343·7-s − 423·8-s + 729·9-s + 1.96e3·11-s − 3.08e3·14-s − 4.89e3·16-s + 6.56e3·18-s + 1.76e4·22-s − 2.27e4·23-s + 1.56e4·25-s − 5.83e3·28-s − 2.12e4·29-s − 1.69e4·32-s + 1.23e4·36-s + 1.01e5·37-s − 1.26e5·43-s + 3.33e4·44-s − 2.04e5·46-s + 1.17e5·49-s + 1.40e5·50-s + 5.03e4·53-s + 1.45e5·56-s − 1.90e5·58-s − 2.50e5·63-s + 1.60e5·64-s + ⋯
L(s)  = 1  + 9/8·2-s + 0.265·4-s − 7-s − 0.826·8-s + 9-s + 1.47·11-s − 9/8·14-s − 1.19·16-s + 9/8·18-s + 1.65·22-s − 1.86·23-s + 25-s − 0.265·28-s − 0.870·29-s − 0.518·32-s + 0.265·36-s + 1.99·37-s − 1.59·43-s + 0.391·44-s − 2.10·46-s + 49-s + 9/8·50-s + 0.338·53-s + 0.826·56-s − 0.978·58-s − 63-s + 0.612·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $1$
Analytic conductor: \(1.61037\)
Root analytic conductor: \(1.26900\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7} (6, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.731183031\)
\(L(\frac12)\) \(\approx\) \(1.731183031\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + p^{3} T \)
good2 \( 1 - 9 T + p^{6} T^{2} \)
3 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
5 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
11 \( 1 - 1962 T + p^{6} T^{2} \)
13 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
17 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
19 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
23 \( 1 + 22734 T + p^{6} T^{2} \)
29 \( 1 + 21222 T + p^{6} T^{2} \)
31 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
37 \( 1 - 101194 T + p^{6} T^{2} \)
41 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
43 \( 1 + 126614 T + p^{6} T^{2} \)
47 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
53 \( 1 - 50346 T + p^{6} T^{2} \)
59 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
61 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
67 \( 1 + 53926 T + p^{6} T^{2} \)
71 \( 1 + 242478 T + p^{6} T^{2} \)
73 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
79 \( 1 - 929378 T + p^{6} T^{2} \)
83 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
89 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
97 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
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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

−21.80948651120245489134036670397, −20.03221185521917167875043032120, −18.42368319527243108721927943531, −16.37109728319191202936161324581, −14.81476474846182295410250461291, −13.33103356152470091311009343780, −12.10892038796338661174385043391, −9.540691988206484661481788512890, −6.42572771132689420056327485309, −3.96607851555454839186798846363, 3.96607851555454839186798846363, 6.42572771132689420056327485309, 9.540691988206484661481788512890, 12.10892038796338661174385043391, 13.33103356152470091311009343780, 14.81476474846182295410250461291, 16.37109728319191202936161324581, 18.42368319527243108721927943531, 20.03221185521917167875043032120, 21.80948651120245489134036670397

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