Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.262 - 0.964i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.260 − 0.260i)2-s + (1.52 + 0.826i)3-s + 1.86i·4-s + (0.611 − 0.180i)6-s + (−0.707 − 0.707i)7-s + (1.00 + 1.00i)8-s + (1.63 + 2.51i)9-s + 3.38i·11-s + (−1.54 + 2.83i)12-s + (−1.59 + 1.59i)13-s − 0.368·14-s − 3.20·16-s + (0.140 − 0.140i)17-s + (1.07 + 0.230i)18-s − 7.34i·19-s + ⋯
L(s)  = 1  + (0.184 − 0.184i)2-s + (0.878 + 0.477i)3-s + 0.932i·4-s + (0.249 − 0.0738i)6-s + (−0.267 − 0.267i)7-s + (0.355 + 0.355i)8-s + (0.544 + 0.838i)9-s + 1.02i·11-s + (−0.445 + 0.819i)12-s + (−0.442 + 0.442i)13-s − 0.0983·14-s − 0.801·16-s + (0.0341 − 0.0341i)17-s + (0.254 + 0.0542i)18-s − 1.68i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.262 - 0.964i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 0.262 - 0.964i)\)
\(L(1)\)  \(\approx\)  \(1.60188 + 1.22397i\)
\(L(\frac12)\)  \(\approx\)  \(1.60188 + 1.22397i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.52 - 0.826i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.260 + 0.260i)T - 2iT^{2} \)
11 \( 1 - 3.38iT - 11T^{2} \)
13 \( 1 + (1.59 - 1.59i)T - 13iT^{2} \)
17 \( 1 + (-0.140 + 0.140i)T - 17iT^{2} \)
19 \( 1 + 7.34iT - 19T^{2} \)
23 \( 1 + (-2.21 - 2.21i)T + 23iT^{2} \)
29 \( 1 - 9.49T + 29T^{2} \)
31 \( 1 - 0.922T + 31T^{2} \)
37 \( 1 + (5.91 + 5.91i)T + 37iT^{2} \)
41 \( 1 + 1.39iT - 41T^{2} \)
43 \( 1 + (0.864 - 0.864i)T - 43iT^{2} \)
47 \( 1 + (-0.651 + 0.651i)T - 47iT^{2} \)
53 \( 1 + (-6.54 - 6.54i)T + 53iT^{2} \)
59 \( 1 - 6.25T + 59T^{2} \)
61 \( 1 - 1.83T + 61T^{2} \)
67 \( 1 + (-0.815 - 0.815i)T + 67iT^{2} \)
71 \( 1 + 9.77iT - 71T^{2} \)
73 \( 1 + (-4.80 + 4.80i)T - 73iT^{2} \)
79 \( 1 + 3.41iT - 79T^{2} \)
83 \( 1 + (6.26 + 6.26i)T + 83iT^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + (-6.71 - 6.71i)T + 97iT^{2} \)
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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

−11.00090352600753239552700332934, −10.06280234361775901644608160716, −9.182123254421966181326615190342, −8.529960291653825950936171402232, −7.33357609979566939444137621019, −6.97718672886883442137499992101, −4.89954699330762970134379811518, −4.30225285818744548658087920135, −3.13634954944933876710125202944, −2.24267840934901454297900504770, 1.10424818071711237209238294349, 2.58344117732893022910998342188, 3.75334307340817935118345949532, 5.16948800877905919655726821337, 6.17095067457567924978640327097, 6.88730648858243544558354509352, 8.157204533206392982733235493040, 8.721751210232845722170214871999, 9.941706321456444967975673364599, 10.31426543039335954225243080737

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