Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.0556 + 0.998i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.79 − 1.79i)2-s + (1.66 + 0.491i)3-s − 4.47i·4-s + (3.87 − 2.10i)6-s + (0.707 + 0.707i)7-s + (−4.45 − 4.45i)8-s + (2.51 + 1.63i)9-s − 1.56i·11-s + (2.19 − 7.43i)12-s + (−2.21 + 2.21i)13-s + 2.54·14-s − 7.09·16-s + (−3.60 + 3.60i)17-s + (7.46 − 1.59i)18-s − 1.68i·19-s + ⋯
L(s)  = 1  + (1.27 − 1.27i)2-s + (0.958 + 0.283i)3-s − 2.23i·4-s + (1.58 − 0.859i)6-s + (0.267 + 0.267i)7-s + (−1.57 − 1.57i)8-s + (0.839 + 0.543i)9-s − 0.472i·11-s + (0.634 − 2.14i)12-s + (−0.615 + 0.615i)13-s + 0.680·14-s − 1.77·16-s + (−0.874 + 0.874i)17-s + (1.75 − 0.375i)18-s − 0.385i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0556 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0556 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.0556 + 0.998i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ -0.0556 + 0.998i)\)
\(L(1)\)  \(\approx\)  \(2.47930 - 2.62144i\)
\(L(\frac12)\)  \(\approx\)  \(2.47930 - 2.62144i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.66 - 0.491i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-1.79 + 1.79i)T - 2iT^{2} \)
11 \( 1 + 1.56iT - 11T^{2} \)
13 \( 1 + (2.21 - 2.21i)T - 13iT^{2} \)
17 \( 1 + (3.60 - 3.60i)T - 17iT^{2} \)
19 \( 1 + 1.68iT - 19T^{2} \)
23 \( 1 + (0.995 + 0.995i)T + 23iT^{2} \)
29 \( 1 + 8.91T + 29T^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
37 \( 1 + (0.440 + 0.440i)T + 37iT^{2} \)
41 \( 1 - 6.44iT - 41T^{2} \)
43 \( 1 + (-5.47 + 5.47i)T - 43iT^{2} \)
47 \( 1 + (3.69 - 3.69i)T - 47iT^{2} \)
53 \( 1 + (2.83 + 2.83i)T + 53iT^{2} \)
59 \( 1 - 5.54T + 59T^{2} \)
61 \( 1 - 7.40T + 61T^{2} \)
67 \( 1 + (-3.75 - 3.75i)T + 67iT^{2} \)
71 \( 1 + 3.61iT - 71T^{2} \)
73 \( 1 + (-5.89 + 5.89i)T - 73iT^{2} \)
79 \( 1 - 17.0iT - 79T^{2} \)
83 \( 1 + (3.21 + 3.21i)T + 83iT^{2} \)
89 \( 1 + 9.40T + 89T^{2} \)
97 \( 1 + (4.39 + 4.39i)T + 97iT^{2} \)
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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

−10.91186864452028858264162544814, −9.909731119263544679408472958723, −9.184161588988439157418781035522, −8.166655200084542862369470346211, −6.76547464320396113994734804231, −5.51927296551270507555089082128, −4.49338562447602383058487622007, −3.79837741585765828705783395073, −2.63663883643888948436510370142, −1.80682524160182366175982669871, 2.37840342285722871610790509396, 3.59686433602717900460649853950, 4.49364329282737723713147060865, 5.44742042741915899023521413011, 6.67309265462540897226133146508, 7.39592016419602226643687728905, 7.928322605925942640315547981488, 8.966502769806212035185318201472, 9.974086618477442087546064479076, 11.44780178732639832061546347058

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