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Properties

Label 2-525-5.3-c2-0-23
Degree $2$
Conductor $525$
Sign $0.850 - 0.525i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.74 + 2.74i)2-s + (1.22 + 1.22i)3-s − 11.0i·4-s − 6.71·6-s + (−1.87 + 1.87i)7-s + (19.3 + 19.3i)8-s + 2.99i·9-s + 10.9·11-s + (13.5 − 13.5i)12-s + (−8.10 − 8.10i)13-s − 10.2i·14-s − 61.7·16-s + (5.51 − 5.51i)17-s + (−8.22 − 8.22i)18-s − 12.1i·19-s + ⋯
L(s)  = 1  + (−1.37 + 1.37i)2-s + (0.408 + 0.408i)3-s − 2.76i·4-s − 1.11·6-s + (−0.267 + 0.267i)7-s + (2.41 + 2.41i)8-s + 0.333i·9-s + 0.993·11-s + (1.12 − 1.12i)12-s + (−0.623 − 0.623i)13-s − 0.732i·14-s − 3.85·16-s + (0.324 − 0.324i)17-s + (−0.457 − 0.457i)18-s − 0.638i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8187814718\)
\(L(\frac12)\) \(\approx\) \(0.8187814718\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
7 \( 1 + (1.87 - 1.87i)T \)
good2 \( 1 + (2.74 - 2.74i)T - 4iT^{2} \)
11 \( 1 - 10.9T + 121T^{2} \)
13 \( 1 + (8.10 + 8.10i)T + 169iT^{2} \)
17 \( 1 + (-5.51 + 5.51i)T - 289iT^{2} \)
19 \( 1 + 12.1iT - 361T^{2} \)
23 \( 1 + (24.3 + 24.3i)T + 529iT^{2} \)
29 \( 1 + 14.8iT - 841T^{2} \)
31 \( 1 + 8.07T + 961T^{2} \)
37 \( 1 + (-34.6 + 34.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 32.0T + 1.68e3T^{2} \)
43 \( 1 + (-13.0 - 13.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-54.1 + 54.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (6.76 + 6.76i)T + 2.80e3iT^{2} \)
59 \( 1 - 44.4iT - 3.48e3T^{2} \)
61 \( 1 + 84.4T + 3.72e3T^{2} \)
67 \( 1 + (0.661 - 0.661i)T - 4.48e3iT^{2} \)
71 \( 1 - 103.T + 5.04e3T^{2} \)
73 \( 1 + (-55.1 - 55.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 68.8iT - 6.24e3T^{2} \)
83 \( 1 + (-71.4 - 71.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 41.6iT - 7.92e3T^{2} \)
97 \( 1 + (25.2 - 25.2i)T - 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.23874119476014551844703184313, −9.514046542927349109981522413163, −8.982459511658300178373222378190, −8.072279415380042655260829698565, −7.31835913570392289997803752977, −6.35122813861850136625579641200, −5.51555899521690741481623177971, −4.32643155276576686944287919329, −2.33618100860218319452204585808, −0.56996938583021205361820702829, 1.14005804231651089112505306275, 2.11000296307911909595353334622, 3.38216674288359725497671884702, 4.17957408978315901317965967213, 6.38981535603952427212487904434, 7.48217665359977215280255882240, 8.029123233332103778884268279579, 9.232809571535136421519897343380, 9.497528118639681595682086576367, 10.45225985534905213860973989404

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