Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{2} $
Sign $-0.229 - 0.973i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)3-s + (1 + i)7-s i·9-s − 2·21-s + (−1 + i)23-s + 2i·29-s + (1 − i)43-s + (−1 − i)47-s + i·49-s + (1 − i)63-s + (1 + i)67-s − 2i·69-s + 81-s + (1 − i)83-s + (−2 − 2i)87-s + ⋯
L(s)  = 1  + (−1 + i)3-s + (1 + i)7-s i·9-s − 2·21-s + (−1 + i)23-s + 2i·29-s + (1 − i)43-s + (−1 − i)47-s + i·49-s + (1 − i)63-s + (1 + i)67-s − 2i·69-s + 81-s + (1 − i)83-s + (−2 − 2i)87-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(800\)    =    \(2^{5} \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.229 - 0.973i$
motivic weight  =  \(0\)
character  :  $\chi_{800} (257, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 800,\ (\ :0),\ -0.229 - 0.973i)$
$L(\frac{1}{2})$  $\approx$  $0.7316422166$
$L(\frac12)$  $\approx$  $0.7316422166$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−10.75221685985421843874095155583, −10.03268865522979731250990617072, −9.099879670111464566614379913831, −8.354335030273642507666476163441, −7.23989387664129141346164886590, −5.94671590199537883366208087967, −5.35118785830813710866323432898, −4.68241429286250276903957052486, −3.53643054532470293052896772089, −1.91087069452013848845759771743, 0.912779280286979310418994085698, 2.16891100401709272297954391108, 4.04655376704796702448425754963, 4.90110252422022193942073224709, 6.05031302786525693150766574450, 6.64426959537030984101543600395, 7.79309233068281090636853498539, 7.987549958995614787489586635325, 9.516285885690007470604571955858, 10.52182806991915867942435966588

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