Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.638 - 0.769i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.393 + 1.68i)3-s + (3.45 − 1.01i)5-s + (−0.403 − 0.349i)7-s + (−2.69 − 1.32i)9-s + (−0.843 + 0.247i)11-s + (2.49 + 3.88i)13-s + (0.352 + 6.22i)15-s + (6.49 + 0.934i)17-s + (−1.01 − 1.17i)19-s + (0.747 − 0.542i)21-s + (−1.87 + 0.854i)23-s + (6.68 − 4.29i)25-s + (3.29 − 4.01i)27-s + 3.54i·29-s + (2.94 − 4.58i)31-s + ⋯
L(s)  = 1  + (−0.227 + 0.973i)3-s + (1.54 − 0.453i)5-s + (−0.152 − 0.132i)7-s + (−0.896 − 0.442i)9-s + (−0.254 + 0.0746i)11-s + (0.692 + 1.07i)13-s + (0.0909 + 1.60i)15-s + (1.57 + 0.226i)17-s + (−0.233 − 0.269i)19-s + (0.163 − 0.118i)21-s + (−0.390 + 0.178i)23-s + (1.33 − 0.859i)25-s + (0.634 − 0.773i)27-s + 0.658i·29-s + (0.529 − 0.824i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.638 - 0.769i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.638 - 0.769i)$
$L(1)$  $\approx$  $1.68547 + 0.791674i$
$L(\frac12)$  $\approx$  $1.68547 + 0.791674i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;67\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.393 - 1.68i)T \)
67 \( 1 + (-4.80 - 6.62i)T \)
good5 \( 1 + (-3.45 + 1.01i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (0.403 + 0.349i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (0.843 - 0.247i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-2.49 - 3.88i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-6.49 - 0.934i)T + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (1.01 + 1.17i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (1.87 - 0.854i)T + (15.0 - 17.3i)T^{2} \)
29 \( 1 - 3.54iT - 29T^{2} \)
31 \( 1 + (-2.94 + 4.58i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 - 6.30T + 37T^{2} \)
41 \( 1 + (1.58 - 11.0i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-4.21 - 0.605i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (8.24 - 3.76i)T + (30.7 - 35.5i)T^{2} \)
53 \( 1 + (0.880 + 6.12i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.431 + 0.671i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (1.65 - 5.64i)T + (-51.3 - 32.9i)T^{2} \)
71 \( 1 + (3.24 - 0.465i)T + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (4.50 + 1.32i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (1.15 + 1.79i)T + (-32.8 + 71.8i)T^{2} \)
83 \( 1 + (0.393 + 1.33i)T + (-69.8 + 44.8i)T^{2} \)
89 \( 1 + (-0.163 - 0.0744i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 + 3.07iT - 97T^{2} \)
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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

−9.953546477823164128174767499660, −9.828022467399469765844615262907, −8.971924109455692968926605366590, −8.071548937968350391153546030101, −6.50626686176829390603393866010, −5.90216176110038163202236357012, −5.10218844796771805174565431898, −4.13132366773891239860693456677, −2.87204846668248972004515545436, −1.43347185545216692535774152563, 1.13565319894448277783931474372, 2.35334420248202265982389163452, 3.25082742751120206157284732557, 5.27138697629791125602214685970, 5.87819288242519428644117882725, 6.41458545636088356764087085098, 7.56202879862628471973119532184, 8.285175278993612161254704898944, 9.388458221747083050912334913841, 10.27938788899755998745116278158

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