Properties

Degree 2
Conductor $ 2^{2} \cdot 29 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 10-s + 11-s + 12-s − 13-s − 15-s + 16-s − 2·19-s − 20-s − 22-s − 24-s + 26-s − 27-s + 29-s + 30-s + 31-s − 32-s + 33-s + 2·38-s − 39-s + 40-s + 43-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 10-s + 11-s + 12-s − 13-s − 15-s + 16-s − 2·19-s − 20-s − 22-s − 24-s + 26-s − 27-s + 29-s + 30-s + 31-s − 32-s + 33-s + 2·38-s − 39-s + 40-s + 43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(116\)    =    \(2^{2} \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{116} (115, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 116,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.4610161263$
$L(\frac12)$  $\approx$  $0.4610161263$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;29\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
29 \( 1 - T \)
good3 \( 1 - T + T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 + T )^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
31 \( 1 - T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 - T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.16016390120846201257187485941, −12.46029134035790053492481655909, −11.68925523873542102048724869802, −10.51454380665689720515485244983, −9.266948877644499009961692823650, −8.471244424587930683412558479394, −7.66013121636265313822893246199, −6.46520354613878822194579118896, −4.06320898332710260335075027351, −2.51709295006250415119800299093, 2.51709295006250415119800299093, 4.06320898332710260335075027351, 6.46520354613878822194579118896, 7.66013121636265313822893246199, 8.471244424587930683412558479394, 9.266948877644499009961692823650, 10.51454380665689720515485244983, 11.68925523873542102048724869802, 12.46029134035790053492481655909, 14.16016390120846201257187485941

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