Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.487 + 0.873i$
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 + (−1.66 + 0.474i)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 + (0.520 + 1.82i)12-s + (4.50 − 4.50i)13-s − 1.75·14-s + 4.99·16-s + (2.13 − 2.13i)17-s + (1.20 − 5.13i)18-s − 4.20i·19-s + ⋯
L(s)  = 1  + (0.879 − 0.879i)2-s + (−0.961 + 0.274i)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 + (0.150 + 0.526i)12-s + (1.25 − 1.25i)13-s − 0.470·14-s + 1.24·16-s + (0.517 − 0.517i)17-s + (0.283 − 1.21i)18-s − 0.965i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.487 + 0.873i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 0.487 + 0.873i)\)
\(L(1)\)  \(\approx\)  \(1.56261 - 0.917030i\)
\(L(\frac12)\)  \(\approx\)  \(1.56261 - 0.917030i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.66 - 0.474i)T \)
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

−10.94978943526494122888069973339, −10.28588240440031842232400422589, −9.295086900483496627952093796483, −7.892033716100451173405122866108, −6.86280352901187162401801669547, −5.64612239571397471515377037653, −4.99114177634517186419095675747, −3.89312572140015764994776280791, −3.01023419226665162862279171851, −1.14875178112633835038721508599, 1.42801144953740884786141924216, 3.66343738877816151076322155057, 4.58739516150629543283883950221, 5.73297517572628144316341972937, 6.19326902238850209861914395055, 6.94050781786056567984247451871, 8.000308880705064743937992500321, 9.158069113640475459832148016215, 10.40164219192003634974653755523, 11.04583069675955723436713218374

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