Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.894 + 0.447i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + i·7-s i·8-s − 2·11-s + i·13-s − 14-s + 16-s i·17-s − 4·19-s − 2i·22-s − 7i·23-s − 26-s i·28-s + 29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s − 0.603·11-s + 0.277i·13-s − 0.267·14-s + 0.250·16-s − 0.242i·17-s − 0.917·19-s − 0.426i·22-s − 1.45i·23-s − 0.196·26-s − 0.188i·28-s + 0.185·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.894 + 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.894 + 0.447i)$
$L(1)$  $\approx$  $1.098415133$
$L(\frac12)$  $\approx$  $1.098415133$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - iT \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 7iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 11iT - 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 - 3iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.441209391517775762363458788091, −8.056352970138622546857262447687, −6.93824515043766526694219211083, −6.51388161801017148728456699899, −5.62442206046540222346720984564, −4.85295102670311579135486305208, −4.16767890302369733025766307495, −2.98695522803573316007910573951, −2.04139695684873212623534625709, −0.38891980352997469944515553764, 1.03404631717384617889362410988, 2.17203488202553299466463835934, 3.08103096442649605749197867140, 3.96838017352575114936592203683, 4.71876505107304421530976423227, 5.62609604427547737567948586981, 6.36490624378062815697104960504, 7.52875966659886035518224209237, 7.917622718233711942763675610896, 8.902186558754098285799163241864

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