Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.816 + 0.577i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 + 0.442i)2-s + 3-s + (1.60 − 1.18i)4-s − 2.27i·5-s + (−1.34 + 0.442i)6-s + 4.97·7-s + (−1.63 + 2.30i)8-s + 9-s + (1.00 + 3.05i)10-s + 2.26·11-s + (1.60 − 1.18i)12-s − 6.26i·13-s + (−6.68 + 2.19i)14-s − 2.27i·15-s + (1.17 − 3.82i)16-s − 0.801·17-s + ⋯
L(s)  = 1  + (−0.949 + 0.312i)2-s + 0.577·3-s + (0.804 − 0.593i)4-s − 1.01i·5-s + (−0.548 + 0.180i)6-s + 1.88·7-s + (−0.578 + 0.815i)8-s + 0.333·9-s + (0.317 + 0.964i)10-s + 0.681·11-s + (0.464 − 0.342i)12-s − 1.73i·13-s + (−1.78 + 0.587i)14-s − 0.586i·15-s + (0.294 − 0.955i)16-s − 0.194·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.816 + 0.577i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.816 + 0.577i)$
$L(1)$  $\approx$  $1.48234 - 0.470939i$
$L(\frac12)$  $\approx$  $1.48234 - 0.470939i$
$L(\frac{3}{2})$   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 \( 1 + (1.34 - 0.442i)T \)
3 \( 1 - T \)
67 \( 1 + (-2.57 - 7.77i)T \)
good5 \( 1 + 2.27iT - 5T^{2} \)
7 \( 1 - 4.97T + 7T^{2} \)
11 \( 1 - 2.26T + 11T^{2} \)
13 \( 1 + 6.26iT - 13T^{2} \)
17 \( 1 + 0.801T + 17T^{2} \)
19 \( 1 - 2.05iT - 19T^{2} \)
23 \( 1 - 7.41iT - 23T^{2} \)
29 \( 1 + 6.30T + 29T^{2} \)
31 \( 1 + 2.55T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 + 0.0427iT - 41T^{2} \)
43 \( 1 + 0.122T + 43T^{2} \)
47 \( 1 + 4.63iT - 47T^{2} \)
53 \( 1 + 6.54iT - 53T^{2} \)
59 \( 1 + 4.77iT - 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
71 \( 1 - 4.26iT - 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + 1.63T + 79T^{2} \)
83 \( 1 - 5.62iT - 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 14.5iT - 97T^{2} \)
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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

−10.00653696868423361117234424117, −9.027749592783952000255396131046, −8.455745681430099437733638254666, −7.891760480140724129547472497861, −7.20468301538286269950581510921, −5.50002548179999429026599328475, −5.17855296941639474522638520202, −3.67585204266017304237741511585, −1.94032026490565861569288622302, −1.14097477755254431610910685647, 1.65832464813359653781923881189, 2.31452064806717166443397962936, 3.73590043768643223782868864101, 4.70114400383709588148409759847, 6.48244115055360805081404032809, 7.09280811744759255286527054820, 7.87696263732245136607113234923, 8.841907690208415296942836683099, 9.198597491058276706912941825563, 10.53291617723460039347169890210

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