Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.729 - 0.684i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.86 + 1.86i)2-s + (0.707 + 0.707i)3-s + 4.93i·4-s + 2.63i·6-s + (2.20 − 1.46i)7-s + (−5.45 + 5.45i)8-s + 1.00i·9-s − 1.46·11-s + (−3.48 + 3.48i)12-s + (−0.887 − 0.887i)13-s + (6.82 + 1.38i)14-s − 10.4·16-s + (2.10 − 2.10i)17-s + (−1.86 + 1.86i)18-s − 3.95·19-s + ⋯
L(s)  = 1  + (1.31 + 1.31i)2-s + (0.408 + 0.408i)3-s + 2.46i·4-s + 1.07i·6-s + (0.833 − 0.552i)7-s + (−1.92 + 1.92i)8-s + 0.333i·9-s − 0.441·11-s + (−1.00 + 1.00i)12-s + (−0.246 − 0.246i)13-s + (1.82 + 0.370i)14-s − 2.61·16-s + (0.510 − 0.510i)17-s + (−0.438 + 0.438i)18-s − 0.908·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.729 - 0.684i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ -0.729 - 0.684i)\)
\(L(1)\)  \(\approx\)  \(1.18996 + 3.00603i\)
\(L(\frac12)\)  \(\approx\)  \(1.18996 + 3.00603i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-2.20 + 1.46i)T \)
good2 \( 1 + (-1.86 - 1.86i)T + 2iT^{2} \)
11 \( 1 + 1.46T + 11T^{2} \)
13 \( 1 + (0.887 + 0.887i)T + 13iT^{2} \)
17 \( 1 + (-2.10 + 2.10i)T - 17iT^{2} \)
19 \( 1 + 3.95T + 19T^{2} \)
23 \( 1 + (-4.13 + 4.13i)T - 23iT^{2} \)
29 \( 1 - 5.18iT - 29T^{2} \)
31 \( 1 + 6.10iT - 31T^{2} \)
37 \( 1 + (2.25 + 2.25i)T + 37iT^{2} \)
41 \( 1 - 0.769iT - 41T^{2} \)
43 \( 1 + (-5.18 + 5.18i)T - 43iT^{2} \)
47 \( 1 + (8.57 - 8.57i)T - 47iT^{2} \)
53 \( 1 + (-0.544 + 0.544i)T - 53iT^{2} \)
59 \( 1 + 3.19T + 59T^{2} \)
61 \( 1 - 1.42iT - 61T^{2} \)
67 \( 1 + (-5.93 - 5.93i)T + 67iT^{2} \)
71 \( 1 - 7.62T + 71T^{2} \)
73 \( 1 + (6.81 + 6.81i)T + 73iT^{2} \)
79 \( 1 - 4.52iT - 79T^{2} \)
83 \( 1 + (-6.75 - 6.75i)T + 83iT^{2} \)
89 \( 1 - 1.19T + 89T^{2} \)
97 \( 1 + (-8.68 + 8.68i)T - 97iT^{2} \)
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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

−11.31871185881789158843415465653, −10.44741313214330463646498440771, −9.032269341410478850038607465046, −8.108361185027061484114103672484, −7.52431571951684636805436287651, −6.60367221272846676570106418964, −5.37223951871500001566605141873, −4.73869308758772080101152630662, −3.86127540145808143562339144044, −2.66489755707349186328857571811, 1.51005958753769707924436565813, 2.44388035684805962288472690505, 3.51532900496239306793152124735, 4.68459316740204918471180544468, 5.45998332660109170263163737304, 6.51021718048477662451446946677, 7.894224721468304255673408084920, 8.944894699556877134415301597708, 9.975032794989957806612151132949, 10.85120963825255929353294649045

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