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Properties

Label 2-525-7.2-c1-0-0
Degree $2$
Conductor $525$
Sign $-0.118 + 0.992i$
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  + (−0.776 + 1.34i)2-s + (0.5 + 0.866i)3-s + (−0.204 − 0.355i)4-s − 1.55·6-s + (−2.60 − 0.478i)7-s − 2.46·8-s + (−0.499 + 0.866i)9-s + (−2.21 − 3.83i)11-s + (0.204 − 0.355i)12-s − 1.73·13-s + (2.66 − 3.12i)14-s + (2.32 − 4.02i)16-s + (−1.36 − 2.36i)17-s + (−0.776 − 1.34i)18-s + (0.152 − 0.264i)19-s + ⋯
L(s)  = 1  + (−0.548 + 0.950i)2-s + (0.288 + 0.499i)3-s + (−0.102 − 0.177i)4-s − 0.633·6-s + (−0.983 − 0.180i)7-s − 0.872·8-s + (−0.166 + 0.288i)9-s + (−0.667 − 1.15i)11-s + (0.0591 − 0.102i)12-s − 0.480·13-s + (0.711 − 0.835i)14-s + (0.581 − 1.00i)16-s + (−0.331 − 0.573i)17-s + (−0.182 − 0.316i)18-s + (0.0350 − 0.0606i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.118 + 0.992i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.118 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0639562 - 0.0720182i\)
\(L(\frac12)\) \(\approx\) \(0.0639562 - 0.0720182i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (2.60 + 0.478i)T \)
good2 \( 1 + (0.776 - 1.34i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (2.21 + 3.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.73T + 13T^{2} \)
17 \( 1 + (1.36 + 2.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.152 + 0.264i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.51 - 6.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.79T + 29T^{2} \)
31 \( 1 + (-2.64 - 4.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.83 - 3.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.71T + 41T^{2} \)
43 \( 1 - 9.71T + 43T^{2} \)
47 \( 1 + (-0.908 + 1.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.857 + 1.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.571 - 0.989i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.77 - 8.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.19 + 7.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + (6.41 + 11.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.35 - 5.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.09T + 83T^{2} \)
89 \( 1 + (2.03 - 3.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.87T + 97T^{2} \)
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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

−11.37301536649559223125073451407, −10.32570981324247574124171537359, −9.470568970985353546949189608578, −8.862042171321727744071377732640, −7.87737468049116625337622815631, −7.16030273674771106064391612234, −6.09853921553533646769399588308, −5.30183488732725081706104096138, −3.64772531163757781052421907379, −2.81474293417123297050899886060, 0.05894336131456572209763637020, 1.99327323852734034382137584623, 2.71114158039884935537272635104, 4.06999311059652771109222045440, 5.68791098143105775883429264988, 6.61530736792496704181316438406, 7.60308331796093946159307682982, 8.688683159292168589442587278454, 9.576699496704525665965394397845, 10.10361322576104016834977184982

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