Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7 $
Sign $-0.845 - 0.533i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.41i)3-s + (−2.41 − 2.41i)5-s + 7-s + (−1.00 + 2.82i)9-s + (−1.82 + 1.82i)11-s + (−1.58 − 1.58i)13-s + (1 − 5.82i)15-s + 6.82i·17-s + (2.41 − 2.41i)19-s + (1 + 1.41i)21-s − 3.65i·23-s + 6.65i·25-s + (−5.00 + 1.41i)27-s + (−3 + 3i)29-s + 10.4i·31-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)3-s + (−1.07 − 1.07i)5-s + 0.377·7-s + (−0.333 + 0.942i)9-s + (−0.551 + 0.551i)11-s + (−0.439 − 0.439i)13-s + (0.258 − 1.50i)15-s + 1.65i·17-s + (0.553 − 0.553i)19-s + (0.218 + 0.308i)21-s − 0.762i·23-s + 1.33i·25-s + (−0.962 + 0.272i)27-s + (−0.557 + 0.557i)29-s + 1.88i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.845 - 0.533i$
motivic weight  =  \(1\)
character  :  $\chi_{1344} (239, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1344,\ (\ :1/2),\ -0.845 - 0.533i)$
$L(1)$  $\approx$  $0.7437242271$
$L(\frac12)$  $\approx$  $0.7437242271$
$L(\frac{3}{2})$   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 \( 1 + (-1 - 1.41i)T \)
7 \( 1 - T \)
good5 \( 1 + (2.41 + 2.41i)T + 5iT^{2} \)
11 \( 1 + (1.82 - 1.82i)T - 11iT^{2} \)
13 \( 1 + (1.58 + 1.58i)T + 13iT^{2} \)
17 \( 1 - 6.82iT - 17T^{2} \)
19 \( 1 + (-2.41 + 2.41i)T - 19iT^{2} \)
23 \( 1 + 3.65iT - 23T^{2} \)
29 \( 1 + (3 - 3i)T - 29iT^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 + (3.82 - 3.82i)T - 37iT^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + (1.82 + 1.82i)T + 43iT^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + (0.171 + 0.171i)T + 53iT^{2} \)
59 \( 1 + (4.07 - 4.07i)T - 59iT^{2} \)
61 \( 1 + (0.414 + 0.414i)T + 61iT^{2} \)
67 \( 1 + (-7 + 7i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (-10.8 - 10.8i)T + 83iT^{2} \)
89 \( 1 + 4.34T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.00839660866598322628309589549, −8.858948613893322074536987979384, −8.410100445067784935103036396420, −7.88731122886830277294680517745, −6.89332091096994920880624372671, −5.17587520473510499240657918423, −4.95860008739282610995746630427, −3.98119724810994481960812186008, −3.14075889838702447559997436709, −1.65867709854020004461908263495, 0.27381194923424441313464692011, 2.08040230191164102626579712332, 3.08062174528199561501755240421, 3.72636817996185279744515505712, 5.06517143850924504364643675346, 6.20295677965717549238787329398, 7.17772531943815417071473098653, 7.58863759015006663191352978647, 8.125481244676711164206579142674, 9.230339890826066631599940103774

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