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  :  $\chi_{804} (41, \cdot )$
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

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

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