Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.713 + 0.700i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.487 − 1.66i)3-s + (1.00 − 1.95i)7-s + (−2.52 + 1.62i)9-s + (4.37 − 5.56i)13-s + (−5.95 + 3.07i)19-s + (−3.74 − 0.720i)21-s + (0.711 − 4.94i)25-s + (3.92 + 3.40i)27-s + (−4.43 − 5.64i)31-s + (4.14 − 7.17i)37-s + (−11.3 − 4.55i)39-s + (−11.4 + 5.24i)43-s + (1.25 + 1.76i)49-s + (8.00 + 8.39i)57-s + (−13.8 + 4.79i)61-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)3-s + (0.380 − 0.738i)7-s + (−0.841 + 0.540i)9-s + (1.21 − 1.54i)13-s + (−1.36 + 0.704i)19-s + (−0.816 − 0.157i)21-s + (0.142 − 0.989i)25-s + (0.755 + 0.654i)27-s + (−0.797 − 1.01i)31-s + (0.680 − 1.17i)37-s + (−1.82 − 0.729i)39-s + (−1.75 + 0.799i)43-s + (0.179 + 0.251i)49-s + (1.06 + 1.11i)57-s + (−1.77 + 0.614i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.713 + 0.700i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (101, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.713 + 0.700i)$
$L(1)$  $\approx$  $0.446702 - 1.09281i$
$L(\frac12)$  $\approx$  $0.446702 - 1.09281i$
$L(\frac{3}{2})$   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 \( 1 + (0.487 + 1.66i)T \)
67 \( 1 + (-6.78 + 4.58i)T \)
good5 \( 1 + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (-1.00 + 1.95i)T + (-4.06 - 5.70i)T^{2} \)
11 \( 1 + (-8.64 - 6.79i)T^{2} \)
13 \( 1 + (-4.37 + 5.56i)T + (-3.06 - 12.6i)T^{2} \)
17 \( 1 + (-16.6 - 3.21i)T^{2} \)
19 \( 1 + (5.95 - 3.07i)T + (11.0 - 15.4i)T^{2} \)
23 \( 1 + (20.4 + 10.5i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.43 + 5.64i)T + (-7.30 + 30.1i)T^{2} \)
37 \( 1 + (-4.14 + 7.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-13.4 - 38.7i)T^{2} \)
43 \( 1 + (11.4 - 5.24i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-2.23 - 46.9i)T^{2} \)
53 \( 1 + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (56.6 - 16.6i)T^{2} \)
61 \( 1 + (13.8 - 4.79i)T + (47.9 - 37.7i)T^{2} \)
71 \( 1 + (-69.7 + 13.4i)T^{2} \)
73 \( 1 + (2.35 + 6.80i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (4.14 + 10.3i)T + (-57.1 + 54.5i)T^{2} \)
83 \( 1 + (-77.0 + 30.8i)T^{2} \)
89 \( 1 + (-74.8 - 48.1i)T^{2} \)
97 \( 1 + (-14.6 - 8.43i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.29789339667156360930333249078, −8.797420128807014252911077259495, −8.038308879633746844456408782279, −7.54234491851588642912970963982, −6.29163452666204653755847595442, −5.84168037254960091667191097914, −4.52164795664883933662762572167, −3.33655200756467069168986318758, −1.91794909297684928661364841755, −0.61259460629678066482681671046, 1.83475340032958111751164529891, 3.31448044225250732684796070614, 4.32548448454276019195451708619, 5.12739183497360670318924340875, 6.15435655134001034876006965266, 6.88266938222018441715782292337, 8.608451363320918983150722777850, 8.724890136131773460800495134182, 9.663966301528957043055432391458, 10.71713849953859582536071509590

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