Properties

Degree 2
Conductor $ 3^{2} \cdot 109 $
Sign $0.577 - 0.816i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.41i·5-s + 7-s + 8-s − 1.41i·10-s − 1.41i·13-s − 14-s − 16-s + 17-s + 1.41i·19-s − 23-s − 1.00·25-s + 1.41i·26-s + 31-s − 34-s + 1.41i·35-s + ⋯
L(s)  = 1  − 2-s + 1.41i·5-s + 7-s + 8-s − 1.41i·10-s − 1.41i·13-s − 14-s − 16-s + 17-s + 1.41i·19-s − 23-s − 1.00·25-s + 1.41i·26-s + 31-s − 34-s + 1.41i·35-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(981\)    =    \(3^{2} \cdot 109\)
\( \varepsilon \)  =  $0.577 - 0.816i$
motivic weight  =  \(0\)
character  :  $\chi_{981} (980, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 981,\ (\ :0),\ 0.577 - 0.816i)$
$L(\frac{1}{2})$  $\approx$  $0.6344526683$
$L(\frac12)$  $\approx$  $0.6344526683$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;109\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;109\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
109 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
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

−10.24501922828186046826561671288, −9.829414406201761837786491810742, −8.348580867267987997166689650584, −7.985199993109958563126994420402, −7.36933289745844762765373199299, −6.17846975473152594450314417416, −5.24561599578540585682920266289, −3.96012904124212978167486029801, −2.84981227279322980961388545654, −1.46120439251126574485432159724, 1.02412040315215052849833468477, 2.02554769498266387721276537475, 4.21736695278699242965459839771, 4.67365549191244132068222878641, 5.60576794923655271891898181577, 7.06457925074114015450698880255, 7.88352989881114887565384654538, 8.603396486529950701470435105101, 9.089094810262748148313373468030, 9.762363584280643289465435771192

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