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Properties

Label 2-496-31.30-c0-0-0
Degree $2$
Conductor $496$
Sign $1$
Analytic cond. $0.247536$
Root an. cond. $0.497530$
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  − 5-s + 7-s + 9-s + 19-s − 31-s − 35-s − 41-s − 45-s − 2·47-s + 59-s + 63-s − 2·67-s + 71-s + 81-s − 95-s − 97-s − 101-s + 103-s + 107-s − 109-s − 113-s + ⋯
L(s)  = 1  − 5-s + 7-s + 9-s + 19-s − 31-s − 35-s − 41-s − 45-s − 2·47-s + 59-s + 63-s − 2·67-s + 71-s + 81-s − 95-s − 97-s − 101-s + 103-s + 107-s − 109-s − 113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(496\)    =    \(2^{4} \cdot 31\)
Sign: $1$
Analytic conductor: \(0.247536\)
Root analytic conductor: \(0.497530\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{496} (433, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 496,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8739657799\)
\(L(\frac12)\) \(\approx\) \(0.8739657799\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 + T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 - T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 + T )^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 - T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 + T )^{2} \)
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 \( 1 + T + T^{2} \)
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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

−11.33979621220307494142617341998, −10.35092728363533814002009073535, −9.400971048775264223122950405864, −8.236781727676124108403907950421, −7.64178938780781159262121842857, −6.82862860558713448622581211010, −5.30668591867096172288592059861, −4.43474446882544503532194136997, −3.44767844354801433253755389737, −1.61544784124339313944370287833, 1.61544784124339313944370287833, 3.44767844354801433253755389737, 4.43474446882544503532194136997, 5.30668591867096172288592059861, 6.82862860558713448622581211010, 7.64178938780781159262121842857, 8.236781727676124108403907950421, 9.400971048775264223122950405864, 10.35092728363533814002009073535, 11.33979621220307494142617341998

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