Properties

Label 2-1216-19.18-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.356183536\)
\(L(\frac12)\) \(\approx\) \(1.356183536\)
\(L(1)\) 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

−10.03427034591441808879612149544, −9.255098424372228179752418993373, −8.172405046070473170772047601897, −7.59999010431101647124163945265, −6.58501746998425739761414145405, −5.62273489322617195952942870189, −4.91269600717961515168772884863, −3.96972632932219871467433158078, −2.38790724946517221009219670095, −1.63850364422660229804002430304, 1.63850364422660229804002430304, 2.38790724946517221009219670095, 3.96972632932219871467433158078, 4.91269600717961515168772884863, 5.62273489322617195952942870189, 6.58501746998425739761414145405, 7.59999010431101647124163945265, 8.172405046070473170772047601897, 9.255098424372228179752418993373, 10.03427034591441808879612149544

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