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Properties

Label 2-416-104.51-c0-0-1
Degree $2$
Conductor $416$
Sign $1$
Analytic cond. $0.207611$
Root an. cond. $0.455643$
Motivic weight $0$
Arithmetic yes
Rational yes
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: $1$
Analytic conductor: \(0.207611\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{416} (207, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.106053865\)
\(L(\frac12)\) \(\approx\) \(1.106053865\)
\(L(1)\) 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

−11.42371445012146906223298386792, −10.10071388774337285274400321869, −9.624135009704371750161913168064, −8.889879139031364702347720607118, −7.86411890344124420829049809431, −6.69492788052190547853235166609, −5.88206968753216184950267243811, −4.48916427945045909880908525906, −3.01462645230284483610556303432, −2.26480681637905692621118531044, 2.26480681637905692621118531044, 3.01462645230284483610556303432, 4.48916427945045909880908525906, 5.88206968753216184950267243811, 6.69492788052190547853235166609, 7.86411890344124420829049809431, 8.889879139031364702347720607118, 9.624135009704371750161913168064, 10.10071388774337285274400321869, 11.42371445012146906223298386792

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