Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.998 + 0.0457i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.24 + 1.24i)2-s + (−0.474 + 1.66i)3-s − 1.09i·4-s + (−1.48 − 2.66i)6-s + (−0.707 − 0.707i)7-s + (−1.12 − 1.12i)8-s + (−2.54 − 1.58i)9-s − 1.55i·11-s + (1.82 + 0.520i)12-s + (4.50 − 4.50i)13-s + 1.75·14-s + 4.99·16-s + (−2.13 + 2.13i)17-s + (5.13 − 1.20i)18-s − 4.20i·19-s + ⋯
L(s)  = 1  + (−0.879 + 0.879i)2-s + (−0.274 + 0.961i)3-s − 0.547i·4-s + (−0.604 − 1.08i)6-s + (−0.267 − 0.267i)7-s + (−0.397 − 0.397i)8-s + (−0.849 − 0.527i)9-s − 0.468i·11-s + (0.526 + 0.150i)12-s + (1.25 − 1.25i)13-s + 0.470·14-s + 1.24·16-s + (−0.517 + 0.517i)17-s + (1.21 − 0.283i)18-s − 0.965i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.998 + 0.0457i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 0.998 + 0.0457i)\)
\(L(1)\)  \(\approx\)  \(0.487523 - 0.0111537i\)
\(L(\frac12)\)  \(\approx\)  \(0.487523 - 0.0111537i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.474 - 1.66i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (1.24 - 1.24i)T - 2iT^{2} \)
11 \( 1 + 1.55iT - 11T^{2} \)
13 \( 1 + (-4.50 + 4.50i)T - 13iT^{2} \)
17 \( 1 + (2.13 - 2.13i)T - 17iT^{2} \)
19 \( 1 + 4.20iT - 19T^{2} \)
23 \( 1 + (3.76 + 3.76i)T + 23iT^{2} \)
29 \( 1 + 2.97T + 29T^{2} \)
31 \( 1 + 5.79T + 31T^{2} \)
37 \( 1 + (-1.23 - 1.23i)T + 37iT^{2} \)
41 \( 1 + 2.68iT - 41T^{2} \)
43 \( 1 + (-2.09 + 2.09i)T - 43iT^{2} \)
47 \( 1 + (0.0358 - 0.0358i)T - 47iT^{2} \)
53 \( 1 + (-4.30 - 4.30i)T + 53iT^{2} \)
59 \( 1 - 4.93T + 59T^{2} \)
61 \( 1 - 3.31T + 61T^{2} \)
67 \( 1 + (1.71 + 1.71i)T + 67iT^{2} \)
71 \( 1 + 5.73iT - 71T^{2} \)
73 \( 1 + (7.26 - 7.26i)T - 73iT^{2} \)
79 \( 1 + 3.59iT - 79T^{2} \)
83 \( 1 + (12.2 + 12.2i)T + 83iT^{2} \)
89 \( 1 - 1.35T + 89T^{2} \)
97 \( 1 + (10.9 + 10.9i)T + 97iT^{2} \)
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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

−10.61792872206581045110676328116, −9.895562173615250149541149469376, −8.781272345398036643074588047799, −8.521613077851563526210872439862, −7.31676431693545759420076497534, −6.19168192771035859344528345464, −5.64823400207173618229312252501, −4.10423440071694293826129021331, −3.17035319762234621974803232153, −0.41265831363990977124167482396, 1.43539914615005691274686843167, 2.26472470141127432062953262300, 3.76822474903103671917030077281, 5.52984331035660836795864538617, 6.34012697397587763065158012291, 7.40101483821293849825141085488, 8.426083229589887033023184889080, 9.135468376918429966060403943719, 9.959530606753676432615165088519, 11.11597287199695651434669817645

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