Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.773 + 0.633i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.71 − 0.246i)3-s + (−1.43 + 4.87i)7-s + (2.87 + 0.845i)9-s + (−5.44 − 4.72i)13-s + (2.19 − 0.645i)19-s + (3.65 − 7.99i)21-s + (3.27 − 3.77i)25-s + (−4.72 − 2.15i)27-s + (−0.512 + 0.444i)31-s − 11.7·37-s + (8.17 + 9.43i)39-s + (−6.94 − 10.8i)43-s + (−15.7 − 10.1i)49-s + (−3.92 + 0.564i)57-s + (−11.7 − 5.34i)61-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)3-s + (−0.540 + 1.84i)7-s + (0.959 + 0.281i)9-s + (−1.51 − 1.30i)13-s + (0.504 − 0.148i)19-s + (0.797 − 1.74i)21-s + (0.654 − 0.755i)25-s + (−0.909 − 0.415i)27-s + (−0.0921 + 0.0798i)31-s − 1.93·37-s + (1.30 + 1.51i)39-s + (−1.05 − 1.64i)43-s + (−2.25 − 1.44i)49-s + (−0.520 + 0.0747i)57-s + (−1.49 − 0.684i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.773 + 0.633i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (137, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.773 + 0.633i)$
$L(1)$  $\approx$  $0.0515266 - 0.144235i$
$L(\frac12)$  $\approx$  $0.0515266 - 0.144235i$
$L(\frac{3}{2})$   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 \( 1 + (1.71 + 0.246i)T \)
67 \( 1 + (3.92 - 7.18i)T \)
good5 \( 1 + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (1.43 - 4.87i)T + (-5.88 - 3.78i)T^{2} \)
11 \( 1 + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (5.44 + 4.72i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (-7.06 + 15.4i)T^{2} \)
19 \( 1 + (-2.19 + 0.645i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (22.0 + 6.47i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (0.512 - 0.444i)T + (4.41 - 30.6i)T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (6.94 + 10.8i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + (45.0 + 13.2i)T^{2} \)
53 \( 1 + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (8.39 - 58.3i)T^{2} \)
61 \( 1 + (11.7 + 5.34i)T + (39.9 + 46.1i)T^{2} \)
71 \( 1 + (-29.4 - 64.5i)T^{2} \)
73 \( 1 + (-6.96 + 15.2i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-10.8 - 9.40i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (54.3 - 62.7i)T^{2} \)
89 \( 1 + (85.3 - 25.0i)T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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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

−10.01842570385802796767034496628, −9.200742395887381214729236877784, −8.231691042805558614176178193438, −7.17690358044657818746319515758, −6.31064374442271675436792945628, −5.30184795939611101480440526332, −5.07902602870690618647235757776, −3.20107933700866768980310599008, −2.15758067506660991869531148657, −0.086910742605287533092897733626, 1.44046715901648137671174425880, 3.41266020022716738275405997006, 4.42447657096343003673931237039, 5.05724864380511012999985624704, 6.48718300256981039787577813812, 7.04116902198539993806515848768, 7.60566417363487954814549924941, 9.276445965574434079569441388253, 9.902348639300758102779075002181, 10.53721849878690525814218916470

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