Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.0640 - 0.997i$
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 + (−0.614 + 0.437i)7-s + (−1.24 + 2.72i)9-s + (−2.32 + 0.563i)13-s + (−4.99 + 7.00i)19-s + (1.21 + 0.485i)21-s + (4.79 + 1.40i)25-s + (5.14 − 0.739i)27-s + (−10.3 − 2.50i)31-s + (5.77 + 9.99i)37-s + (2.99 + 2.85i)39-s + (1.51 + 1.31i)43-s + (−2.10 + 6.07i)49-s + (14.8 + 0.708i)57-s + (8.44 + 10.7i)61-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)3-s + (−0.232 + 0.165i)7-s + (−0.415 + 0.909i)9-s + (−0.644 + 0.156i)13-s + (−1.14 + 1.60i)19-s + (0.264 + 0.105i)21-s + (0.959 + 0.281i)25-s + (0.989 − 0.142i)27-s + (−1.85 − 0.450i)31-s + (0.949 + 1.64i)37-s + (0.479 + 0.457i)39-s + (0.230 + 0.199i)43-s + (−0.300 + 0.868i)49-s + (1.97 + 0.0939i)57-s + (1.08 + 1.37i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0640 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0640 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.0640 - 0.997i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (653, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.0640 - 0.997i)$
$L(1)$  $\approx$  $0.441986 + 0.414534i$
$L(\frac12)$  $\approx$  $0.441986 + 0.414534i$
$L(\frac{3}{2})$   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 \( 1 + (-6.98 + 4.26i)T \)
good5 \( 1 + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (0.614 - 0.437i)T + (2.28 - 6.61i)T^{2} \)
11 \( 1 + (2.59 + 10.6i)T^{2} \)
13 \( 1 + (2.32 - 0.563i)T + (11.5 - 5.95i)T^{2} \)
17 \( 1 + (-15.7 - 6.31i)T^{2} \)
19 \( 1 + (4.99 - 7.00i)T + (-6.21 - 17.9i)T^{2} \)
23 \( 1 + (-13.3 - 18.7i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (10.3 + 2.50i)T + (27.5 + 14.2i)T^{2} \)
37 \( 1 + (-5.77 - 9.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-32.2 + 25.3i)T^{2} \)
43 \( 1 + (-1.51 - 1.31i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + (46.7 - 4.46i)T^{2} \)
53 \( 1 + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-8.44 - 10.7i)T + (-14.3 + 59.2i)T^{2} \)
71 \( 1 + (-65.9 + 26.3i)T^{2} \)
73 \( 1 + (11.3 - 8.92i)T + (17.2 - 70.9i)T^{2} \)
79 \( 1 + (5.50 + 5.76i)T + (-3.75 + 78.9i)T^{2} \)
83 \( 1 + (-60.0 + 57.2i)T^{2} \)
89 \( 1 + (-36.9 - 80.9i)T^{2} \)
97 \( 1 + (16.2 - 9.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.57242455401370440708211463804, −9.699086396942774294914477638672, −8.604732802956549009853649648072, −7.81675807692426371617979553958, −6.95902543157275869183588191085, −6.13710084977143335451486610525, −5.34766441466020182162892696536, −4.17887252520140737904935155674, −2.71262242905281526801542513329, −1.52849314858197882712409510626, 0.31867768979834763096175439973, 2.49653745046140697829124511188, 3.73611951405331577046802332411, 4.68475596310806171751904670303, 5.46264236495963395132771295460, 6.55754841986778328085081486725, 7.27977890744195939004764188643, 8.676656950527541331820373641063, 9.244486190644114307655654041269, 10.13179914848237623514111928007

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