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 − 6·11-s + i·13-s − 14-s + 16-s − 3i·17-s + 4·19-s − 6i·22-s − 3i·23-s − 26-s i·28-s + 3·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s − 1.80·11-s + 0.277i·13-s − 0.267·14-s + 0.250·16-s − 0.727i·17-s + 0.917·19-s − 1.27i·22-s − 0.625i·23-s − 0.196·26-s − 0.188i·28-s + 0.557·29-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  =  $(2,\ 3150,\ (\ :1/2),\ 0.894 - 0.447i)$
$L(1)$  $\approx$  $1.345981977$
$L(\frac12)$  $\approx$  $1.345981977$
$L(\frac{3}{2})$   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 \( 1 + 6T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - 9T + 59T^{2} \)
61 \( 1 - 11T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.484416722006998457196752843412, −8.028024437350136986872283569691, −7.23548241246269076465242685135, −6.59662262362520316785158338603, −5.44546515561877733406635975891, −5.24929416754664928317637140974, −4.26477065713028948207898379877, −3.05783996250439134404285734861, −2.30977461324972319335074126966, −0.58985917366547297451182552514, 0.801377077126200488323559083344, 2.05084966473532951062737151288, 3.01192924951030813575281827571, 3.66196077915573327863344971750, 4.92710562915008985118549109562, 5.22043629582227355792129521072, 6.32047192516003649063087768939, 7.28176097123705725341777601123, 8.149724297132105758874691392591, 8.396890690561103216066728692385

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