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 − 7i·13-s + 14-s + 16-s + 7i·17-s − 8·19-s − 2i·22-s − 5i·23-s − 7·26-s i·28-s + 9·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.377i·7-s + 0.353i·8-s + 0.603·11-s − 1.94i·13-s + 0.267·14-s + 0.250·16-s + 1.69i·17-s − 1.83·19-s − 0.426i·22-s − 1.04i·23-s − 1.37·26-s − 0.188i·28-s + 1.67·29-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  =  $(2,\ 3150,\ (\ :1/2),\ -0.894 + 0.447i)$
$L(1)$  $\approx$  $1.076347863$
$L(\frac12)$  $\approx$  $1.076347863$
$L(\frac{3}{2})$   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 \( 1 + 7iT - 13T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 + 5iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 11T + 41T^{2} \)
43 \( 1 + 3iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 3iT - 53T^{2} \)
59 \( 1 - 7T + 59T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
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

−8.340033520648519684455229869048, −8.101684733450376567932985661289, −6.61436164253623202977620184795, −6.14202258076991458656565764398, −5.19463959774859308182043047669, −4.32464660042979915556858373212, −3.51648091045218432085558216112, −2.62588015808960914988669655533, −1.67568919804336700782523855238, −0.34920238595241510993526294456, 1.26326178822355029715871554354, 2.46935528906122070996103266726, 3.76511834172431116782694093027, 4.47297915487857364706196279247, 5.04970164333507718777609582642, 6.32957002532125918216990431556, 6.73277316906065924708901819067, 7.25166980335507169927531130144, 8.359063473281505591935829106303, 8.853275170807142537891153385401

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