Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $-0.261 + 0.965i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.17 − 1.27i)3-s + (−2.19 + 3.79i)5-s + (−0.5 − 0.866i)7-s + (−0.253 − 2.98i)9-s + (−2.69 − 4.66i)11-s + (1.27 − 2.20i)13-s + (2.27 + 7.23i)15-s − 2.58·17-s + 6.72·19-s + (−1.69 − 0.377i)21-s + (−0.400 + 0.693i)23-s + (−7.09 − 12.2i)25-s + (−4.10 − 3.18i)27-s + (−1.87 − 3.24i)29-s + (1.69 − 2.93i)31-s + ⋯
L(s)  = 1  + (0.676 − 0.736i)3-s + (−0.979 + 1.69i)5-s + (−0.188 − 0.327i)7-s + (−0.0843 − 0.996i)9-s + (−0.811 − 1.40i)11-s + (0.352 − 0.610i)13-s + (0.586 + 1.86i)15-s − 0.625·17-s + 1.54·19-s + (−0.368 − 0.0823i)21-s + (−0.0834 + 0.144i)23-s + (−1.41 − 2.45i)25-s + (−0.790 − 0.612i)27-s + (−0.347 − 0.601i)29-s + (0.304 − 0.527i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.261 + 0.965i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -0.261 + 0.965i)\)
\(L(1)\)  \(\approx\)  \(1.144878145\)
\(L(\frac12)\)  \(\approx\)  \(1.144878145\)
\(L(\frac{3}{2})\)   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 \( 1 \)
3 \( 1 + (-1.17 + 1.27i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (2.19 - 3.79i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.69 + 4.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.27 + 2.20i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.58T + 17T^{2} \)
19 \( 1 - 6.72T + 19T^{2} \)
23 \( 1 + (0.400 - 0.693i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.87 + 3.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.69 + 2.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.38T + 37T^{2} \)
41 \( 1 + (-3.19 + 5.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.381 + 0.661i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.13 + 7.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.94T + 53T^{2} \)
59 \( 1 + (-2.78 + 4.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.14 - 7.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.946 - 1.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.34T + 71T^{2} \)
73 \( 1 + 8.65T + 73T^{2} \)
79 \( 1 + (6.64 + 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.86 - 3.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.99T + 89T^{2} \)
97 \( 1 + (1.48 + 2.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.832639536577868309953276460710, −8.594150236254741900021665612570, −7.79930393405590288349668133950, −7.44285269278333958835445209681, −6.50500247890692729140314175511, −5.74196270113433242657688497357, −3.86848390508924113478945668033, −3.21696094288303624487013342666, −2.57422703179546698489388327359, −0.48334758071664142102092831429, 1.59312683230284723648540780305, 3.04722589507455425534693724351, 4.27819902713521196060327028478, 4.70735902538626324595435863327, 5.49030556252917327516983927748, 7.21031476794292864660667540337, 7.926449066575445216307393500415, 8.595807211899749335591400245826, 9.424261531520257830172222581105, 9.753866176357295549062421350362

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