Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.744 - 0.667i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 + 0.390i)2-s + (1.69 + 1.06i)4-s − 1.82i·5-s + 7-s + (1.88 + 2.10i)8-s + (0.714 − 2.48i)10-s + 3.75i·11-s + 3.09i·13-s + (1.35 + 0.390i)14-s + (1.74 + 3.59i)16-s + 3.61·17-s + 1.65i·19-s + (1.94 − 3.10i)20-s + (−1.46 + 5.10i)22-s − 7.47·23-s + ⋯
L(s)  = 1  + (0.961 + 0.276i)2-s + (0.847 + 0.531i)4-s − 0.818i·5-s + 0.377·7-s + (0.667 + 0.744i)8-s + (0.226 − 0.786i)10-s + 1.13i·11-s + 0.859i·13-s + (0.363 + 0.104i)14-s + (0.436 + 0.899i)16-s + 0.875·17-s + 0.379i·19-s + (0.434 − 0.693i)20-s + (−0.313 + 1.08i)22-s − 1.55·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.744 - 0.667i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ 0.744 - 0.667i)\)
\(L(1)\)  \(\approx\)  \(3.393865409\)
\(L(\frac12)\)  \(\approx\)  \(3.393865409\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.35 - 0.390i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 1.82iT - 5T^{2} \)
11 \( 1 - 3.75iT - 11T^{2} \)
13 \( 1 - 3.09iT - 13T^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 - 1.65iT - 19T^{2} \)
23 \( 1 + 7.47T + 23T^{2} \)
29 \( 1 + 2.38iT - 29T^{2} \)
31 \( 1 - 8.97T + 31T^{2} \)
37 \( 1 + 8.94iT - 37T^{2} \)
41 \( 1 - 3.04T + 41T^{2} \)
43 \( 1 - 1.82iT - 43T^{2} \)
47 \( 1 - 6.21T + 47T^{2} \)
53 \( 1 + 3.21iT - 53T^{2} \)
59 \( 1 + 12.4iT - 59T^{2} \)
61 \( 1 - 9.91iT - 61T^{2} \)
67 \( 1 - 8.73iT - 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 + 7.24T + 79T^{2} \)
83 \( 1 + 8.99iT - 83T^{2} \)
89 \( 1 + 1.54T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−9.621848835172427967221224731722, −8.544249914996658421347661622904, −7.84483555978409806146261462779, −7.12035519369554541589543092776, −6.12947111719952337995153475496, −5.35813530048269940145763362319, −4.43051794573833968725621963162, −4.02935776920591965836393888340, −2.49606039039722994638017005945, −1.52303549331520617163047751093, 1.09877910647828018382567152149, 2.65186646841676043569685531443, 3.20235014724597723415054436474, 4.21491364389691861658898857995, 5.30084897576594460063766327362, 5.99794655412238371111617114691, 6.69575824385735699276827547623, 7.73409077632760441470646339434, 8.334758560360336585092396245268, 9.696926682277524950074011891367

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