Properties

Degree $2$
Conductor $416$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 7-s + 13-s − 15-s − 17-s + 21-s − 27-s − 2·31-s − 35-s − 37-s + 39-s + 43-s + 47-s − 51-s − 65-s + 71-s − 81-s + 85-s + 91-s − 2·93-s − 105-s − 2·107-s − 109-s − 111-s + 2·113-s − 119-s + ⋯
L(s)  = 1  + 3-s − 5-s + 7-s + 13-s − 15-s − 17-s + 21-s − 27-s − 2·31-s − 35-s − 37-s + 39-s + 43-s + 47-s − 51-s − 65-s + 71-s − 81-s + 85-s + 91-s − 2·93-s − 105-s − 2·107-s − 109-s − 111-s + 2·113-s − 119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{416} (207, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024488922\)
\(L(\frac12)\) \(\approx\) \(1.024488922\)
\(L(1)\) 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 \( 1 - T \)
good3 \( 1 - T + T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 + T )^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 - T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−11.21920871375829792749765569162, −10.88130085406947463401502402381, −9.250450978319346720755416548801, −8.585287277419569846017455910415, −7.954143502560078217961115903988, −7.10955820197510546801346318054, −5.61729941197096691827675041220, −4.26984902056982014947676898937, −3.47895634233078360386963741748, −1.99755916622639326258291511234, 1.99755916622639326258291511234, 3.47895634233078360386963741748, 4.26984902056982014947676898937, 5.61729941197096691827675041220, 7.10955820197510546801346318054, 7.954143502560078217961115903988, 8.585287277419569846017455910415, 9.250450978319346720755416548801, 10.88130085406947463401502402381, 11.21920871375829792749765569162

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