Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.791 + 0.610i$
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 + (0.518 − 0.543i)7-s + (2.87 − 0.845i)9-s + (−1.21 − 6.27i)13-s + (3.30 − 3.15i)19-s + (0.755 − 1.06i)21-s + (3.27 + 3.77i)25-s + (4.72 − 2.15i)27-s + (−2.02 + 10.5i)31-s + (0.656 − 1.13i)37-s + (−3.62 − 10.4i)39-s + (5.92 − 9.21i)43-s + (0.306 + 6.42i)49-s + (4.89 − 6.21i)57-s + (−0.304 + 3.19i)61-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)3-s + (0.196 − 0.205i)7-s + (0.959 − 0.281i)9-s + (−0.335 − 1.74i)13-s + (0.758 − 0.723i)19-s + (0.164 − 0.231i)21-s + (0.654 + 0.755i)25-s + (0.909 − 0.415i)27-s + (−0.364 + 1.89i)31-s + (0.107 − 0.186i)37-s + (−0.580 − 1.67i)39-s + (0.903 − 1.40i)43-s + (0.0437 + 0.918i)49-s + (0.647 − 0.823i)57-s + (−0.0390 + 0.408i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.791 + 0.610i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (233, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.791 + 0.610i)$
$L(1)$  $\approx$  $2.12756 - 0.725345i$
$L(\frac12)$  $\approx$  $2.12756 - 0.725345i$
$L(\frac{3}{2})$   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 \( 1 + (0.979 - 8.12i)T \)
good5 \( 1 + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (-0.518 + 0.543i)T + (-0.333 - 6.99i)T^{2} \)
11 \( 1 + (10.8 - 2.08i)T^{2} \)
13 \( 1 + (1.21 + 6.27i)T + (-12.0 + 4.83i)T^{2} \)
17 \( 1 + (-9.86 + 13.8i)T^{2} \)
19 \( 1 + (-3.30 + 3.15i)T + (0.904 - 18.9i)T^{2} \)
23 \( 1 + (-16.6 - 15.8i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.02 - 10.5i)T + (-28.7 - 11.5i)T^{2} \)
37 \( 1 + (-0.656 + 1.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-40.8 - 3.89i)T^{2} \)
43 \( 1 + (-5.92 + 9.21i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-11.0 + 45.6i)T^{2} \)
53 \( 1 + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.304 - 3.19i)T + (-59.8 - 11.5i)T^{2} \)
71 \( 1 + (-41.1 - 57.8i)T^{2} \)
73 \( 1 + (17.0 + 1.62i)T + (71.6 + 13.8i)T^{2} \)
79 \( 1 + (16.7 + 5.81i)T + (62.0 + 48.8i)T^{2} \)
83 \( 1 + (27.1 - 78.4i)T^{2} \)
89 \( 1 + (85.3 + 25.0i)T^{2} \)
97 \( 1 + (-6.63 - 3.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.22405244610748237458099372983, −9.144044483574236052862487607717, −8.554294195967066206610315211896, −7.49708833014385231565862081402, −7.14293196174974522550591499962, −5.65435514551167864072742004579, −4.73691750741240007753925365857, −3.42228688081936835748418737820, −2.72296002989334334985270124806, −1.14791170346109205893118592776, 1.68144095028887655586289540288, 2.70774525137604317808925274675, 3.98215804350479812475780018999, 4.67908959155014717402847138396, 6.04512766113216269667864558103, 7.09004605444775458832261677957, 7.84169433575529828507466225946, 8.736753448671618930713853251499, 9.463914570673657498078566202937, 10.03963343783695428507723936266

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