Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.949 + 0.314i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.936 − 1.45i)3-s + (−0.502 + 5.26i)7-s + (−1.24 − 2.72i)9-s + (4.62 − 4.85i)13-s + (4.83 − 0.461i)19-s + (7.20 + 5.66i)21-s + (4.79 − 1.40i)25-s + (−5.14 − 0.739i)27-s + (7.68 + 8.05i)31-s + (3.82 + 6.61i)37-s + (−2.73 − 11.2i)39-s + (4.02 − 3.48i)43-s + (−20.6 − 3.97i)49-s + (3.85 − 7.47i)57-s + (−4.67 − 11.6i)61-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)3-s + (−0.190 + 1.99i)7-s + (−0.415 − 0.909i)9-s + (1.28 − 1.34i)13-s + (1.10 − 0.105i)19-s + (1.57 + 1.23i)21-s + (0.959 − 0.281i)25-s + (−0.989 − 0.142i)27-s + (1.38 + 1.44i)31-s + (0.628 + 1.08i)37-s + (−0.438 − 1.80i)39-s + (0.613 − 0.531i)43-s + (−2.94 − 0.567i)49-s + (0.510 − 0.990i)57-s + (−0.598 − 1.49i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.949 + 0.314i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (353, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.949 + 0.314i)$
$L(1)$  $\approx$  $1.90946 - 0.307578i$
$L(\frac12)$  $\approx$  $1.90946 - 0.307578i$
$L(\frac{3}{2})$   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 \( 1 + (-0.936 + 1.45i)T \)
67 \( 1 + (-3.56 + 7.36i)T \)
good5 \( 1 + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (0.502 - 5.26i)T + (-6.87 - 1.32i)T^{2} \)
11 \( 1 + (7.96 + 7.59i)T^{2} \)
13 \( 1 + (-4.62 + 4.85i)T + (-0.618 - 12.9i)T^{2} \)
17 \( 1 + (13.3 + 10.5i)T^{2} \)
19 \( 1 + (-4.83 + 0.461i)T + (18.6 - 3.59i)T^{2} \)
23 \( 1 + (22.8 + 2.18i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.68 - 8.05i)T + (-1.47 + 30.9i)T^{2} \)
37 \( 1 + (-3.82 - 6.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (38.0 - 15.2i)T^{2} \)
43 \( 1 + (-4.02 + 3.48i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 + (-27.2 + 38.2i)T^{2} \)
53 \( 1 + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (4.67 + 11.6i)T + (-44.1 + 42.0i)T^{2} \)
71 \( 1 + (55.8 - 43.8i)T^{2} \)
73 \( 1 + (15.2 - 6.10i)T + (52.8 - 50.3i)T^{2} \)
79 \( 1 + (14.8 + 3.60i)T + (70.2 + 36.1i)T^{2} \)
83 \( 1 + (-19.5 + 80.6i)T^{2} \)
89 \( 1 + (-36.9 + 80.9i)T^{2} \)
97 \( 1 + (11.5 - 6.66i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.06919838035854200039746897648, −9.046196875616530680186522943842, −8.522541942516521253914882705118, −7.918159747517547913206293396285, −6.60361760129787531470620216913, −5.94107666963932576660629022342, −5.12510585308966806303415003244, −3.18615257778677203421282450754, −2.77143063072658290528788283898, −1.25927592708598596762209839966, 1.21301817204718841709713289958, 3.02804534073877247561669024329, 4.12128796947432618383429084398, 4.37038463499058573766970910855, 5.91534971724208152206274785656, 7.02158055254361737619526178313, 7.71595087418418680260802965661, 8.729886575074537345962170781044, 9.581710769461863285729440660938, 10.20576713426313346301491724710

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