Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.194 - 0.980i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.41i)3-s − 2·5-s + (3.94 + 2.28i)7-s + (−1.00 + 2.82i)9-s + (−1.72 + 2.98i)11-s + (4.5 − 2.59i)13-s + (−2 − 2.82i)15-s + (2.17 − 1.25i)17-s + (−1.5 − 2.59i)19-s + (0.724 + 7.86i)21-s + (−8.17 + 4.71i)23-s − 25-s + (−5.00 + 1.41i)27-s + (7.62 + 4.40i)29-s + (−1.5 − 0.866i)31-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)3-s − 0.894·5-s + (1.49 + 0.861i)7-s + (−0.333 + 0.942i)9-s + (−0.520 + 0.900i)11-s + (1.24 − 0.720i)13-s + (−0.516 − 0.730i)15-s + (0.527 − 0.304i)17-s + (−0.344 − 0.596i)19-s + (0.158 + 1.71i)21-s + (−1.70 + 0.984i)23-s − 0.200·25-s + (−0.962 + 0.272i)27-s + (1.41 + 0.817i)29-s + (−0.269 − 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.194 - 0.980i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (365, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.194 - 0.980i)$
$L(1)$  $\approx$  $1.11660 + 1.35919i$
$L(\frac12)$  $\approx$  $1.11660 + 1.35919i$
$L(\frac{3}{2})$   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 \( 1 + (-1 - 1.41i)T \)
67 \( 1 + (-8 - 1.73i)T \)
good5 \( 1 + 2T + 5T^{2} \)
7 \( 1 + (-3.94 - 2.28i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.72 - 2.98i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.17 + 1.25i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.17 - 4.71i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.62 - 4.40i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0505 + 0.0874i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.724 + 1.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 12.5iT - 43T^{2} \)
47 \( 1 + (2.72 + 1.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 6.29iT - 59T^{2} \)
61 \( 1 + (0.398 - 0.230i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (-3.82 - 2.20i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.94 + 3.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.39 + 5.42i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.07 + 4.08i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 16.0iT - 89T^{2} \)
97 \( 1 + (2.84 - 1.64i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.51813824591413610796329953085, −9.652637625143854450468510491701, −8.520992176267362542142757040823, −8.146149268498349776926868601450, −7.52704863177309229765839636218, −5.81394908606086645678508820606, −4.95319984068026430352370400162, −4.21355485292172450076571675996, −3.10452491063081726081917585052, −1.85586136177714784575576231631, 0.874407742149730763175277766597, 2.08872438393129253203191730676, 3.72743960242981800705260923880, 4.19633567698243252583469471942, 5.74917165277786606635523765439, 6.69619877314121659852713035107, 7.80691328418299928860344364564, 8.213904348530514521097646089770, 8.589074636065075732005220362414, 10.22285723760322320130437277464

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