Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.573 - 0.818i$
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 + (1.75 + 1.97i)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 + (0.665 + 0.746i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.573 - 0.818i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 0.573 - 0.818i)\)
\(L(1)\)  \(\approx\)  \(0.610323 + 0.317574i\)
\(L(\frac12)\)  \(\approx\)  \(0.610323 + 0.317574i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.75 - 1.97i)T \)
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

−10.85999027120737780912174275500, −10.23623724936758086700808320570, −9.205386786721565454144941990100, −8.412511083296935421771753896471, −8.208570714435245661922665291280, −6.51929904161647337910472134225, −5.20371998715590683977569873459, −3.98840523407866027567680350669, −2.59784488178061145577188308081, −1.81364517846332089508471187078, 0.52639777040599529367534198123, 2.32349815409581057291478826051, 4.07775702637598138076895183949, 5.39918644486421150834214254472, 6.59076241621676685842661192229, 7.27212383108605225646133128266, 8.047222960717724225068044409178, 8.633168910311181553412785771271, 9.551203080053097063470599182150, 10.54384285938212567408878003732

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