Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 7 $
Sign $-0.978 + 0.204i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.621 − 1.27i)2-s + (−1.22 − 1.57i)4-s − 3.69i·5-s + 7-s + (−2.76 + 0.578i)8-s + (−4.69 − 2.29i)10-s − 3.21i·11-s + 5.08i·13-s + (0.621 − 1.27i)14-s + (−0.985 + 3.87i)16-s − 0.616·17-s − 4.48i·19-s + (−5.83 + 4.54i)20-s + (−4.08 − 1.99i)22-s − 1.38·23-s + ⋯
L(s)  = 1  + (0.439 − 0.898i)2-s + (−0.613 − 0.789i)4-s − 1.65i·5-s + 0.377·7-s + (−0.978 + 0.204i)8-s + (−1.48 − 0.726i)10-s − 0.968i·11-s + 1.40i·13-s + (0.166 − 0.339i)14-s + (−0.246 + 0.969i)16-s − 0.149·17-s − 1.02i·19-s + (−1.30 + 1.01i)20-s + (−0.870 − 0.425i)22-s − 0.288·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.978 + 0.204i$
motivic weight  =  \(1\)
character  :  $\chi_{504} (253, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 504,\ (\ :1/2),\ -0.978 + 0.204i)$
$L(1)$  $\approx$  $0.156425 - 1.51270i$
$L(\frac12)$  $\approx$  $0.156425 - 1.51270i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.621 + 1.27i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 3.69iT - 5T^{2} \)
11 \( 1 + 3.21iT - 11T^{2} \)
13 \( 1 - 5.08iT - 13T^{2} \)
17 \( 1 + 0.616T + 17T^{2} \)
19 \( 1 + 4.48iT - 19T^{2} \)
23 \( 1 + 1.38T + 23T^{2} \)
29 \( 1 - 5.67iT - 29T^{2} \)
31 \( 1 - 6.91T + 31T^{2} \)
37 \( 1 + 6.91iT - 37T^{2} \)
41 \( 1 - 0.616T + 41T^{2} \)
43 \( 1 + 7.99iT - 43T^{2} \)
47 \( 1 + 4.97T + 47T^{2} \)
53 \( 1 + 4.48iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 + 9.56iT - 67T^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 + 5.23T + 79T^{2} \)
83 \( 1 - 10.4iT - 83T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 - 9.73T + 97T^{2} \)
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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

−10.75235833946735871311148511221, −9.438268541877214491347410474816, −8.947481214843817276315468589444, −8.257581945502088935952201976037, −6.57444226657235613926318276715, −5.35386426367311937921771459798, −4.71117536544338130968831507296, −3.78375436680447551412573481508, −2.08762099808199299939956451658, −0.820140150956060062796269478836, 2.58118396245048387723952940934, 3.59059082676602374985579561688, 4.80161758993674977462207546031, 6.02395745706690985302037867393, 6.65312077090968522550278510762, 7.76870776500293467006418625528, 8.051061963274187492921425888447, 9.765860166808184914598897563275, 10.27568631411612037310694132671, 11.39873316371642237501285584837

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