Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.767 - 0.640i$
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 + (−0.842 + 4.37i)7-s + (1.96 − 2.26i)9-s + (7.17 − 0.341i)13-s + (−6.35 + 1.22i)19-s + (−1.81 − 7.49i)21-s + (−4.20 + 2.70i)25-s + (−1.46 + 4.98i)27-s + (−4.75 − 0.226i)31-s + (−6.07 + 10.5i)37-s + (−11.0 + 5.69i)39-s + (5.51 + 0.792i)43-s + (−11.9 − 4.76i)49-s + (9.13 − 6.50i)57-s + (5.09 + 5.34i)61-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)3-s + (−0.318 + 1.65i)7-s + (0.654 − 0.755i)9-s + (1.98 − 0.0947i)13-s + (−1.45 + 0.281i)19-s + (−0.396 − 1.63i)21-s + (−0.841 + 0.540i)25-s + (−0.281 + 0.959i)27-s + (−0.854 − 0.0406i)31-s + (−0.999 + 1.73i)37-s + (−1.77 + 0.912i)39-s + (0.840 + 0.120i)43-s + (−1.70 − 0.680i)49-s + (1.20 − 0.861i)57-s + (0.652 + 0.684i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.767 - 0.640i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (413, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.767 - 0.640i)$
$L(1)$  $\approx$  $0.269942 + 0.744785i$
$L(\frac12)$  $\approx$  $0.269942 + 0.744785i$
$L(\frac{3}{2})$   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 \( 1 + (7.90 + 2.12i)T \)
good5 \( 1 + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (0.842 - 4.37i)T + (-6.49 - 2.60i)T^{2} \)
11 \( 1 + (0.523 + 10.9i)T^{2} \)
13 \( 1 + (-7.17 + 0.341i)T + (12.9 - 1.23i)T^{2} \)
17 \( 1 + (-4.00 - 16.5i)T^{2} \)
19 \( 1 + (6.35 - 1.22i)T + (17.6 - 7.06i)T^{2} \)
23 \( 1 + (-22.5 - 4.35i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.75 + 0.226i)T + (30.8 + 2.94i)T^{2} \)
37 \( 1 + (6.07 - 10.5i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (29.6 - 28.2i)T^{2} \)
43 \( 1 + (-5.51 - 0.792i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (15.3 + 44.4i)T^{2} \)
53 \( 1 + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (-5.09 - 5.34i)T + (-2.90 + 60.9i)T^{2} \)
71 \( 1 + (-16.7 + 68.9i)T^{2} \)
73 \( 1 + (8.03 - 7.66i)T + (3.47 - 72.9i)T^{2} \)
79 \( 1 + (1.45 - 2.81i)T + (-45.8 - 64.3i)T^{2} \)
83 \( 1 + (73.7 + 38.0i)T^{2} \)
89 \( 1 + (58.2 + 67.2i)T^{2} \)
97 \( 1 + (10.7 + 6.23i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.73169660725300485083243253214, −9.772855303415433411661551415669, −8.855105885609157836506734795738, −8.376795937062091686930100731589, −6.78920384809115383224247816711, −5.92716422375825769237497415885, −5.64130805666523370196298395397, −4.29398163444083567303063240562, −3.27675535890525095642973436111, −1.69654422135277408042758808645, 0.45396738612953854568685746704, 1.73630593151091948153853871440, 3.76987704371770525803980637687, 4.29400227973402290554919934166, 5.73500994555094249346581878905, 6.45821802683072240823239219653, 7.16589796640455591699973582617, 8.048692428590624580412794833360, 9.080919172015534771888082399153, 10.40713700891934390194643837005

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