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Properties

Label 2-1216-19.18-c0-0-0
Degree $2$
Conductor $1216$
Sign $1$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
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 + 11-s − 17-s − 19-s + 2·23-s − 35-s + 43-s + 45-s − 47-s + 55-s + 61-s − 63-s − 73-s − 77-s + 81-s − 2·83-s − 85-s − 95-s + 99-s − 2·101-s + 2·115-s + 119-s + ⋯
L(s)  = 1  + 5-s − 7-s + 9-s + 11-s − 17-s − 19-s + 2·23-s − 35-s + 43-s + 45-s − 47-s + 55-s + 61-s − 63-s − 73-s − 77-s + 81-s − 2·83-s − 85-s − 95-s + 99-s − 2·101-s + 2·115-s + 119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $1$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1216} (1025, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.240850096\)
\(L(\frac12)\) \(\approx\) \(1.240850096\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 - T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T + T^{2} \)
23 \( ( 1 - T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
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 \( 1 - T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 + T )^{2} \)
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

−9.743925553197831389853782382945, −9.312766806520324350449380745313, −8.577246503382308181212911404506, −7.07199785023897052651582584541, −6.68470538831733392691825038165, −5.94153595478288606476463090847, −4.74386377035284362674949760855, −3.85812125665234711937576948696, −2.63373583294376273584783891474, −1.46331533509913767217191803252, 1.46331533509913767217191803252, 2.63373583294376273584783891474, 3.85812125665234711937576948696, 4.74386377035284362674949760855, 5.94153595478288606476463090847, 6.68470538831733392691825038165, 7.07199785023897052651582584541, 8.577246503382308181212911404506, 9.312766806520324350449380745313, 9.743925553197831389853782382945

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