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Properties

Label 2-525-105.23-c1-0-10
Degree $2$
Conductor $525$
Sign $-0.328 - 0.944i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + (1.10 + 1.33i)3-s + (−2 − 1.41i)6-s + (1.15 − 2.38i)7-s + (1.99 − 2i)8-s + (−0.548 + 2.94i)9-s + (−3.67 + 2.12i)11-s + (3.53 + 3.53i)13-s + (−0.707 + 3.67i)14-s + (−1.99 + 3.46i)16-s + (0.366 − 1.36i)17-s + (−0.330 − 4.22i)18-s + (6.06 + 3.5i)19-s + (4.44 − 1.09i)21-s + (4.24 − 4.24i)22-s + (1.09 + 4.09i)23-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.639 + 0.769i)3-s + (−0.816 − 0.577i)6-s + (0.436 − 0.899i)7-s + (0.707 − 0.707i)8-s + (−0.182 + 0.983i)9-s + (−1.10 + 0.639i)11-s + (0.980 + 0.980i)13-s + (−0.188 + 0.981i)14-s + (−0.499 + 0.866i)16-s + (0.0887 − 0.331i)17-s + (−0.0779 − 0.996i)18-s + (1.39 + 0.802i)19-s + (0.970 − 0.239i)21-s + (0.904 − 0.904i)22-s + (0.228 + 0.854i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.328 - 0.944i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.328 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.553176 + 0.778262i\)
\(L(\frac12)\) \(\approx\) \(0.553176 + 0.778262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.10 - 1.33i)T \)
5 \( 1 \)
7 \( 1 + (-1.15 + 2.38i)T \)
good2 \( 1 + (1.36 - 0.366i)T + (1.73 - i)T^{2} \)
11 \( 1 + (3.67 - 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.53 - 3.53i)T + 13iT^{2} \)
17 \( 1 + (-0.366 + 1.36i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.06 - 3.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.09 - 4.09i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.258 - 0.965i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + (3.53 + 3.53i)T + 43iT^{2} \)
47 \( 1 + (-5.46 + 1.46i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.73 - 0.732i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.965 - 0.258i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 + (1.81 - 6.76i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.06 - 3.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1 - i)T - 83iT^{2} \)
89 \( 1 + (-7.77 + 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.07 + 7.07i)T - 97iT^{2} \)
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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

−10.75252473965823955520652573142, −9.962851242302929703323725588415, −9.445842604239466547534111220451, −8.483118351687080807321771683768, −7.67768951054325542866930670160, −7.19397108154637068774526936663, −5.37337991657660994854530039715, −4.33001600641972095253018475536, −3.46503762032844108718405780733, −1.58880674238501735875086698575, 0.78350872324225690542582429530, 2.20029195779545056747025100039, 3.25229871183858720922471779681, 5.16837315211030117478676147553, 5.94290582944479096602560979460, 7.51009631782801173015381292225, 8.086326080579669598155356686763, 8.764019824171973730671964702339, 9.399614474959777094575728998306, 10.60926734033297252446002643207

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