Properties

Degree 2
Conductor $ 2^{2} \cdot 89 $
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 + 2·7-s + 8-s − 10-s − 12-s + 2·14-s + 15-s + 16-s − 17-s − 19-s − 20-s − 2·21-s − 23-s − 24-s + 27-s + 2·28-s + 30-s − 31-s + 32-s − 34-s − 2·35-s − 38-s − 40-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s − 10-s − 12-s + 2·14-s + 15-s + 16-s − 17-s − 19-s − 20-s − 2·21-s − 23-s − 24-s + 27-s + 2·28-s + 30-s − 31-s + 32-s − 34-s − 2·35-s − 38-s − 40-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(356\)    =    \(2^{2} \cdot 89\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{356} (355, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 356,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.058354802$
$L(\frac12)$  $\approx$  $1.058354802$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−11.67664387409176254961069404999, −11.12641832666094511225923344933, −10.59630604998005499570730013389, −8.486875419216600821463090171005, −7.81465077644172741318499151707, −6.72180728980827205506652916631, −5.57538027644770783049218703171, −4.72657548264463000766072728667, −4.01600802562592568972815723771, −2.00491779343604680931144269321, 2.00491779343604680931144269321, 4.01600802562592568972815723771, 4.72657548264463000766072728667, 5.57538027644770783049218703171, 6.72180728980827205506652916631, 7.81465077644172741318499151707, 8.486875419216600821463090171005, 10.59630604998005499570730013389, 11.12641832666094511225923344933, 11.67664387409176254961069404999

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