Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.869 + 0.493i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)3-s + (−1.73 + i)4-s + (−0.189 + 2.63i)7-s + (−0.866 − 0.499i)9-s + (−3 − 5.19i)11-s + (0.517 + 1.93i)12-s + (−1.41 − 1.41i)13-s + (1.99 − 3.46i)16-s + (−5.79 − 1.55i)17-s + (1.73 − 3i)19-s + (2.50 + 0.866i)21-s + (−0.707 + 0.707i)27-s + (−2.31 − 4.76i)28-s + (−4.5 + 2.59i)31-s + (−5.79 + 1.55i)33-s + ⋯
L(s)  = 1  + (0.149 − 0.557i)3-s + (−0.866 + 0.5i)4-s + (−0.0716 + 0.997i)7-s + (−0.288 − 0.166i)9-s + (−0.904 − 1.56i)11-s + (0.149 + 0.557i)12-s + (−0.392 − 0.392i)13-s + (0.499 − 0.866i)16-s + (−1.40 − 0.376i)17-s + (0.397 − 0.688i)19-s + (0.545 + 0.188i)21-s + (−0.136 + 0.136i)27-s + (−0.436 − 0.899i)28-s + (−0.808 + 0.466i)31-s + (−1.00 + 0.270i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.869 + 0.493i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (82, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 525,\ (\ :1/2),\ -0.869 + 0.493i)$
$L(1)$  $\approx$  $0.0972919 - 0.368494i$
$L(\frac12)$  $\approx$  $0.0972919 - 0.368494i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (0.189 - 2.63i)T \)
good2 \( 1 + (1.73 - i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.41 + 1.41i)T + 13iT^{2} \)
17 \( 1 + (5.79 + 1.55i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.73 + 3i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.36 - 2.24i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + (3.67 - 3.67i)T - 43iT^{2} \)
47 \( 1 + (-1.55 - 5.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-10.0 - 2.68i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.19 + 9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.5 - 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.896 + 3.34i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (0.258 - 0.965i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.33 - 2.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.24 - 4.24i)T + 83iT^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.19 + 9.19i)T - 97iT^{2} \)
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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

−10.56720873123200665021651506301, −9.169108917748294228708556781394, −8.755702631909762633491719190126, −8.008799109155946575969478701413, −6.94722092468280410647730619143, −5.66432371992523535466428848062, −4.99391307161124015396356665960, −3.38845253616131406567043228967, −2.52048140508014136373237582391, −0.21278174363786797361557722989, 1.97719339403107622471735206980, 3.82753760399692837185009775914, 4.55516690837679482890276387157, 5.31768732458663500760834348411, 6.76176298683531198647303714524, 7.67213420367201754409428041763, 8.741260632140540763223049176116, 9.676868167030146226691463184908, 10.20494373566625270967073722547, 10.83057350732314291491317656424

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