Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.475 + 0.879i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.48 − 1.48i)2-s + (−0.707 − 0.707i)3-s + 2.43i·4-s + 2.10i·6-s + (1.97 + 1.75i)7-s + (0.640 − 0.640i)8-s + 1.00i·9-s − 2.67·11-s + (1.71 − 1.71i)12-s + (−1.22 − 1.22i)13-s + (−0.320 − 5.55i)14-s + 2.95·16-s + (4.74 − 4.74i)17-s + (1.48 − 1.48i)18-s + 6.01·19-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)2-s + (−0.408 − 0.408i)3-s + 1.21i·4-s + 0.859i·6-s + (0.746 + 0.665i)7-s + (0.226 − 0.226i)8-s + 0.333i·9-s − 0.805·11-s + (0.496 − 0.496i)12-s + (−0.340 − 0.340i)13-s + (−0.0857 − 1.48i)14-s + 0.738·16-s + (1.15 − 1.15i)17-s + (0.350 − 0.350i)18-s + 1.38·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.475 + 0.879i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ -0.475 + 0.879i)\)
\(L(1)\)  \(\approx\)  \(0.359713 - 0.603596i\)
\(L(\frac12)\)  \(\approx\)  \(0.359713 - 0.603596i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.97 - 1.75i)T \)
good2 \( 1 + (1.48 + 1.48i)T + 2iT^{2} \)
11 \( 1 + 2.67T + 11T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + 13iT^{2} \)
17 \( 1 + (-4.74 + 4.74i)T - 17iT^{2} \)
19 \( 1 - 6.01T + 19T^{2} \)
23 \( 1 + (-0.175 + 0.175i)T - 23iT^{2} \)
29 \( 1 + 0.304iT - 29T^{2} \)
31 \( 1 + 7.25iT - 31T^{2} \)
37 \( 1 + (-0.735 - 0.735i)T + 37iT^{2} \)
41 \( 1 + 7.05iT - 41T^{2} \)
43 \( 1 + (0.304 - 0.304i)T - 43iT^{2} \)
47 \( 1 + (0.556 - 0.556i)T - 47iT^{2} \)
53 \( 1 + (-4.99 + 4.99i)T - 53iT^{2} \)
59 \( 1 + 7.98T + 59T^{2} \)
61 \( 1 - 5.53iT - 61T^{2} \)
67 \( 1 + (-3.43 - 3.43i)T + 67iT^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (10.0 + 10.0i)T + 73iT^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + (-4.88 - 4.88i)T + 83iT^{2} \)
89 \( 1 - 6.91T + 89T^{2} \)
97 \( 1 + (-8.84 + 8.84i)T - 97iT^{2} \)
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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

−10.54857327805998943830806126629, −9.801468442314386082241775300721, −9.009812032211000729028222764987, −7.84723404328687255363233143057, −7.54202526592518008546258287397, −5.71275792628490904664925035430, −5.08365315488498037347685415652, −3.11480294355767398465593827997, −2.16065759741835446792252677336, −0.75529894950362805676002048427, 1.18126719075770289462981361181, 3.46823414492189519842872875971, 4.92568589484887623021907810723, 5.69946679756483845307410427738, 6.84085214113904244807073783186, 7.70431234817752288077636446559, 8.225009920266372660923181781863, 9.370233380990491860267232727137, 10.14852902112952612932973004473, 10.71593885366891333019886751949

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