Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.893 + 0.449i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.540 − 0.540i)2-s + (0.707 + 0.707i)3-s − 1.41i·4-s − 0.763i·6-s + (−0.614 − 2.57i)7-s + (−1.84 + 1.84i)8-s + 1.00i·9-s − 3.85·11-s + (1.00 − 1.00i)12-s + (−3.66 − 3.66i)13-s + (−1.05 + 1.72i)14-s − 0.839·16-s + (1.49 − 1.49i)17-s + (0.540 − 0.540i)18-s − 0.0697·19-s + ⋯
L(s)  = 1  + (−0.381 − 0.381i)2-s + (0.408 + 0.408i)3-s − 0.708i·4-s − 0.311i·6-s + (−0.232 − 0.972i)7-s + (−0.652 + 0.652i)8-s + 0.333i·9-s − 1.16·11-s + (0.289 − 0.289i)12-s + (−1.01 − 1.01i)13-s + (−0.282 + 0.460i)14-s − 0.209·16-s + (0.361 − 0.361i)17-s + (0.127 − 0.127i)18-s − 0.0160·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.893 + 0.449i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ -0.893 + 0.449i)\)
\(L(1)\)  \(\approx\)  \(0.169362 - 0.713593i\)
\(L(\frac12)\)  \(\approx\)  \(0.169362 - 0.713593i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.614 + 2.57i)T \)
good2 \( 1 + (0.540 + 0.540i)T + 2iT^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 + (3.66 + 3.66i)T + 13iT^{2} \)
17 \( 1 + (-1.49 + 1.49i)T - 17iT^{2} \)
19 \( 1 + 0.0697T + 19T^{2} \)
23 \( 1 + (-0.534 + 0.534i)T - 23iT^{2} \)
29 \( 1 - 2.77iT - 29T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 + (6.18 + 6.18i)T + 37iT^{2} \)
41 \( 1 + 8.68iT - 41T^{2} \)
43 \( 1 + (-2.77 + 2.77i)T - 43iT^{2} \)
47 \( 1 + (-5.49 + 5.49i)T - 47iT^{2} \)
53 \( 1 + (6.13 - 6.13i)T - 53iT^{2} \)
59 \( 1 - 6.97T + 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 + (0.416 + 0.416i)T + 67iT^{2} \)
71 \( 1 + 8.12T + 71T^{2} \)
73 \( 1 + (-9.55 - 9.55i)T + 73iT^{2} \)
79 \( 1 + 9.86iT - 79T^{2} \)
83 \( 1 + (-1.63 - 1.63i)T + 83iT^{2} \)
89 \( 1 - 5.05T + 89T^{2} \)
97 \( 1 + (-6.85 + 6.85i)T - 97iT^{2} \)
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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

−10.53560192584384023759879905621, −9.830986648000032498507837854124, −8.934459456671552021779307621062, −7.83805383626239139884602615556, −7.07317928390833299798743724983, −5.54156003654932878073388786961, −4.90752501310725860241126144925, −3.39760962491729295211258333955, −2.29897617923040026717457080403, −0.43163873589456642552152614344, 2.27509065146973482334440938494, 3.14390535056540980099145753063, 4.62603702029558319344013962875, 5.93008194544740765160164532854, 6.91636862321964361778547626075, 7.76801873888989334650698460126, 8.422690371513840938492421701784, 9.300570489846346653458903929018, 9.994140662348653285634607132395, 11.48454422245188661200726314954

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