Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.974 + 0.226i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.54 − 1.54i)2-s + (0.00622 − 1.73i)3-s − 2.76i·4-s + (−2.66 − 2.68i)6-s + (−0.707 − 0.707i)7-s + (−1.18 − 1.18i)8-s + (−2.99 − 0.0215i)9-s − 3.38i·11-s + (−4.79 − 0.0172i)12-s + (0.206 − 0.206i)13-s − 2.18·14-s + 1.87·16-s + (0.167 − 0.167i)17-s + (−4.66 + 4.59i)18-s + 5.31i·19-s + ⋯
L(s)  = 1  + (1.09 − 1.09i)2-s + (0.00359 − 0.999i)3-s − 1.38i·4-s + (−1.08 − 1.09i)6-s + (−0.267 − 0.267i)7-s + (−0.419 − 0.419i)8-s + (−0.999 − 0.00718i)9-s − 1.02i·11-s + (−1.38 − 0.00497i)12-s + (0.0573 − 0.0573i)13-s − 0.583·14-s + 0.467·16-s + (0.0406 − 0.0406i)17-s + (−1.09 + 1.08i)18-s + 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.974 + 0.226i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ -0.974 + 0.226i)\)
\(L(1)\)  \(\approx\)  \(0.267229 - 2.33157i\)
\(L(\frac12)\)  \(\approx\)  \(0.267229 - 2.33157i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.00622 + 1.73i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-1.54 + 1.54i)T - 2iT^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
13 \( 1 + (-0.206 + 0.206i)T - 13iT^{2} \)
17 \( 1 + (-0.167 + 0.167i)T - 17iT^{2} \)
19 \( 1 - 5.31iT - 19T^{2} \)
23 \( 1 + (5.07 + 5.07i)T + 23iT^{2} \)
29 \( 1 - 2.84T + 29T^{2} \)
31 \( 1 - 9.11T + 31T^{2} \)
37 \( 1 + (-5.27 - 5.27i)T + 37iT^{2} \)
41 \( 1 + 0.0314iT - 41T^{2} \)
43 \( 1 + (-3.76 + 3.76i)T - 43iT^{2} \)
47 \( 1 + (3.56 - 3.56i)T - 47iT^{2} \)
53 \( 1 + (-3.55 - 3.55i)T + 53iT^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 6.80T + 61T^{2} \)
67 \( 1 + (6.34 + 6.34i)T + 67iT^{2} \)
71 \( 1 + 3.95iT - 71T^{2} \)
73 \( 1 + (8.61 - 8.61i)T - 73iT^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 + (-3.88 - 3.88i)T + 83iT^{2} \)
89 \( 1 + 2.00T + 89T^{2} \)
97 \( 1 + (2.26 + 2.26i)T + 97iT^{2} \)
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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

−10.70883613578561284228302330099, −10.02188102840301327608851097506, −8.503791580840762188877134111219, −7.82723972417123568816308460653, −6.34508892205911946673683637587, −5.84786399948305781335664422142, −4.49226652796811192078236495353, −3.35843452269923362069780345377, −2.44973905426380321912758243258, −1.05105456908498128972700640593, 2.75659835611311532162840821443, 4.03023853341238543833630535431, 4.69074796680255937374250984992, 5.59717464077009707639147983393, 6.45207159413305737924751540564, 7.44622544722364631525097082272, 8.450270106603706938426228935701, 9.564174603538052893773475170880, 10.19116950805962601000870762906, 11.46547871821930801281051209564

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