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Properties

Label 2-525-35.17-c1-0-0
Degree $2$
Conductor $525$
Sign $0.632 + 0.774i$
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  + (−0.602 + 2.24i)2-s + (−0.965 + 0.258i)3-s + (−2.95 − 1.70i)4-s − 2.32i·6-s + (−0.519 + 2.59i)7-s + (2.33 − 2.33i)8-s + (0.866 − 0.499i)9-s + (−1.76 + 3.05i)11-s + (3.30 + 0.884i)12-s + (−4.49 − 4.49i)13-s + (−5.51 − 2.73i)14-s + (0.421 + 0.729i)16-s + (−0.481 − 1.79i)17-s + (0.602 + 2.24i)18-s + (−0.0699 − 0.121i)19-s + ⋯
L(s)  = 1  + (−0.425 + 1.58i)2-s + (−0.557 + 0.149i)3-s + (−1.47 − 0.854i)4-s − 0.950i·6-s + (−0.196 + 0.980i)7-s + (0.824 − 0.824i)8-s + (0.288 − 0.166i)9-s + (−0.531 + 0.921i)11-s + (0.952 + 0.255i)12-s + (−1.24 − 1.24i)13-s + (−1.47 − 0.730i)14-s + (0.105 + 0.182i)16-s + (−0.116 − 0.436i)17-s + (0.141 + 0.529i)18-s + (−0.0160 − 0.0277i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.632 + 0.774i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.632 + 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0328833 - 0.0156095i\)
\(L(\frac12)\) \(\approx\) \(0.0328833 - 0.0156095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (0.519 - 2.59i)T \)
good2 \( 1 + (0.602 - 2.24i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (1.76 - 3.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.49 + 4.49i)T + 13iT^{2} \)
17 \( 1 + (0.481 + 1.79i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.0699 + 0.121i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.72 - 0.997i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.01iT - 29T^{2} \)
31 \( 1 + (4.56 + 2.63i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.50 + 5.61i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.903iT - 41T^{2} \)
43 \( 1 + (2.38 - 2.38i)T - 43iT^{2} \)
47 \( 1 + (-2.38 - 0.639i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.726 - 2.71i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.15 + 5.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.69 - 5.01i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 - 2.77i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.09T + 71T^{2} \)
73 \( 1 + (9.04 - 2.42i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.30 + 4.21i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.37 + 7.37i)T + 83iT^{2} \)
89 \( 1 + (-1.75 - 3.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.70 + 8.70i)T - 97iT^{2} \)
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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

−11.66300154121908782541798159808, −10.36607471525950508846766951369, −9.600774024824907904346122716397, −8.908957605971510413587071774180, −7.67272587018670124580944462314, −7.28752641355119266957302671351, −6.08141518271932545500941750910, −5.35626527765462838411724703336, −4.73352683726843793643920878750, −2.65264305133121572911365701303, 0.02651617499517446680549578952, 1.49240016588786687625419890358, 2.87825740456416332643924725783, 4.02687678364175155472233384132, 4.98786934075529794974778852499, 6.51879917616039786468418717176, 7.45652951291669278028062010597, 8.681069342093551072195124097425, 9.537830721318577849500238352187, 10.41744965527695503485359790535

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