Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $-0.997 + 0.0754i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 + 2.95i)5-s + (−2.27 + 1.35i)7-s + (3.01 + 5.22i)11-s + 0.417·13-s + (1.27 + 2.20i)17-s + (−1.74 + 3.01i)19-s + (4.54 − 7.87i)23-s + (−3.33 − 5.78i)25-s − 1.67·29-s + (−3.82 − 6.62i)31-s + (−0.110 − 9.04i)35-s + (−4.69 + 8.12i)37-s − 2.13·41-s − 6.03·43-s + (−3.94 + 6.84i)47-s + ⋯
L(s)  = 1  + (−0.764 + 1.32i)5-s + (−0.859 + 0.510i)7-s + (0.909 + 1.57i)11-s + 0.115·13-s + (0.309 + 0.535i)17-s + (−0.399 + 0.692i)19-s + (0.948 − 1.64i)23-s + (−0.667 − 1.15i)25-s − 0.311·29-s + (−0.686 − 1.18i)31-s + (−0.0186 − 1.52i)35-s + (−0.771 + 1.33i)37-s − 0.333·41-s − 0.919·43-s + (−0.576 + 0.997i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.997 + 0.0754i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (1297, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ -0.997 + 0.0754i)\)
\(L(1)\)  \(\approx\)  \(0.7666526242\)
\(L(\frac12)\)  \(\approx\)  \(0.7666526242\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7 \( 1 + (2.27 - 1.35i)T \)
good5 \( 1 + (1.70 - 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.01 - 5.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.417T + 13T^{2} \)
17 \( 1 + (-1.27 - 2.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.74 - 3.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.54 + 7.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.67T + 29T^{2} \)
31 \( 1 + (3.82 + 6.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.69 - 8.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.13T + 41T^{2} \)
43 \( 1 + 6.03T + 43T^{2} \)
47 \( 1 + (3.94 - 6.84i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.967 - 1.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.91 + 6.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.67 - 8.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.516 + 0.894i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.650T + 71T^{2} \)
73 \( 1 + (0.642 + 1.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.20 + 14.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + (6.70 - 11.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.26T + 97T^{2} \)
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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

−9.966210793340796187980942725346, −9.186446438919193146334680420062, −8.206096922587023090563178350337, −7.30549043773621489806169524096, −6.63558067242039008842139542591, −6.19228460847789443245176313679, −4.70949495759329256009222956678, −3.80307485592881464319147233018, −3.03442903562607318736737500387, −1.93777651563857625830458734048, 0.32535476866388621563466925221, 1.29979023278873495490275978908, 3.39161517323223663160203696338, 3.67109553359806975733135796343, 4.91815902473951686437601672596, 5.63708302107148017822701076547, 6.73526188255207378114343788445, 7.43842585598348171886623817995, 8.508683512704512797353338870992, 8.984553830507561920490776492577

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