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Properties

Label 2-525-35.12-c1-0-20
Degree $2$
Conductor $525$
Sign $-0.604 + 0.796i$
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  + (0.394 − 0.105i)2-s + (0.258 − 0.965i)3-s + (−1.58 + 0.916i)4-s − 0.408i·6-s + (0.605 − 2.57i)7-s + (−1.10 + 1.10i)8-s + (−0.866 − 0.499i)9-s + (−0.463 − 0.803i)11-s + (0.474 + 1.77i)12-s + (−4.08 − 4.08i)13-s + (−0.0332 − 1.08i)14-s + (1.51 − 2.62i)16-s + (0.719 + 0.192i)17-s + (−0.394 − 0.105i)18-s + (−1.21 + 2.11i)19-s + ⋯
L(s)  = 1  + (0.278 − 0.0747i)2-s + (0.149 − 0.557i)3-s + (−0.793 + 0.458i)4-s − 0.166i·6-s + (0.228 − 0.973i)7-s + (−0.391 + 0.391i)8-s + (−0.288 − 0.166i)9-s + (−0.139 − 0.242i)11-s + (0.136 + 0.511i)12-s + (−1.13 − 1.13i)13-s + (−0.00889 − 0.288i)14-s + (0.378 − 0.655i)16-s + (0.174 + 0.0467i)17-s + (−0.0929 − 0.0249i)18-s + (−0.279 + 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.604 + 0.796i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.604 + 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.429561 - 0.865616i\)
\(L(\frac12)\) \(\approx\) \(0.429561 - 0.865616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-0.605 + 2.57i)T \)
good2 \( 1 + (-0.394 + 0.105i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.463 + 0.803i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.08 + 4.08i)T + 13iT^{2} \)
17 \( 1 + (-0.719 - 0.192i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.21 - 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.34 + 5.00i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 8.08iT - 29T^{2} \)
31 \( 1 + (1.05 - 0.607i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.76 - 0.472i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.97iT - 41T^{2} \)
43 \( 1 + (-0.781 + 0.781i)T - 43iT^{2} \)
47 \( 1 + (2.70 + 10.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.42 - 1.72i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.91 - 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.72 + 2.15i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.80 - 10.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 + (1.07 - 4.02i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.02 - 4.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.91 - 5.91i)T + 83iT^{2} \)
89 \( 1 + (-7.78 + 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.89 - 4.89i)T - 97iT^{2} \)
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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

−10.40057039442824609332630986098, −9.807335819822930939104192707221, −8.445187011028889775863789482948, −7.952575638200795369505233666114, −7.10764822636764828252859062527, −5.78979381933271240574332195706, −4.74518222150734488611283953855, −3.75153862169568335171493942769, −2.56050722747333954086898629305, −0.50775443487843324216176837797, 2.06543061012536682549789798802, 3.55764387453970344177956967707, 4.79803822905942727869297338867, 5.22286393588812896702862060180, 6.40113553928694539820731118719, 7.64671200652146331003772928857, 8.948235462454714882411591265110, 9.239947601594628600905224327946, 10.09231844875532067284903193321, 11.14665425532080215178974355313

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