Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.766 + 0.642i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.57 − 0.719i)3-s + (3.27 − 1.13i)7-s + (1.96 − 2.26i)9-s + (−0.172 + 0.334i)13-s + (0.0563 − 0.162i)19-s + (4.34 − 4.14i)21-s + (−4.20 + 2.70i)25-s + (1.46 − 4.98i)27-s + (0.658 + 1.27i)31-s + (−2.33 − 4.04i)37-s + (−0.0309 + 0.650i)39-s + (8.39 + 1.20i)43-s + (3.94 − 3.10i)49-s + (−0.0283 − 0.297i)57-s + (5.40 − 1.31i)61-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)3-s + (1.23 − 0.428i)7-s + (0.654 − 0.755i)9-s + (−0.0477 + 0.0926i)13-s + (0.0129 − 0.0373i)19-s + (0.948 − 0.904i)21-s + (−0.841 + 0.540i)25-s + (0.281 − 0.959i)27-s + (0.118 + 0.229i)31-s + (−0.383 − 0.664i)37-s + (−0.00496 + 0.104i)39-s + (1.28 + 0.184i)43-s + (0.563 − 0.442i)49-s + (−0.00375 − 0.0393i)57-s + (0.692 − 0.168i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.766 + 0.642i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (185, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.766 + 0.642i)$
$L(1)$  $\approx$  $2.24508 - 0.816854i$
$L(\frac12)$  $\approx$  $2.24508 - 0.816854i$
$L(\frac{3}{2})$   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 \( 1 + (-1.57 + 0.719i)T \)
67 \( 1 + (1.34 - 8.07i)T \)
good5 \( 1 + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-3.27 + 1.13i)T + (5.50 - 4.32i)T^{2} \)
11 \( 1 + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (0.172 - 0.334i)T + (-7.54 - 10.5i)T^{2} \)
17 \( 1 + (-12.3 + 11.7i)T^{2} \)
19 \( 1 + (-0.0563 + 0.162i)T + (-14.9 - 11.7i)T^{2} \)
23 \( 1 + (7.52 + 21.7i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.658 - 1.27i)T + (-17.9 + 25.2i)T^{2} \)
37 \( 1 + (2.33 + 4.04i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.66 + 39.8i)T^{2} \)
43 \( 1 + (-8.39 - 1.20i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-46.1 - 8.89i)T^{2} \)
53 \( 1 + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (-5.40 + 1.31i)T + (54.2 - 27.9i)T^{2} \)
71 \( 1 + (-51.3 - 48.9i)T^{2} \)
73 \( 1 + (0.548 + 2.25i)T + (-64.8 + 33.4i)T^{2} \)
79 \( 1 + (17.2 - 0.819i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (-3.94 - 82.9i)T^{2} \)
89 \( 1 + (58.2 + 67.2i)T^{2} \)
97 \( 1 + (2.91 - 1.68i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.08354786636980386629006968903, −9.167678675037266907845974818381, −8.378423198005916273447696600203, −7.65129558521832462979739930047, −7.03470041147916157622831730024, −5.76416892419867881930227987746, −4.57786519341220448956712969161, −3.70508244963826976398290990756, −2.36386591405024418678036366116, −1.29699180805480834515896753256, 1.69798548666627572894048408576, 2.71307173481384239532601518367, 4.00791488732623348523365384738, 4.82075815612165292299306135121, 5.78689540088319594989970373836, 7.17918720593765059257957496855, 8.040338493646982629331743839867, 8.535783745707281776832388825917, 9.442998368241401031886900076871, 10.26141937374317566503032486361

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