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Properties

Label 2-525-5.2-c2-0-29
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  + (−0.992 − 0.992i)2-s + (−1.22 + 1.22i)3-s − 2.02i·4-s + 2.43·6-s + (−1.87 − 1.87i)7-s + (−5.98 + 5.98i)8-s − 2.99i·9-s + 6.89·11-s + (2.48 + 2.48i)12-s + (11.8 − 11.8i)13-s + 3.71i·14-s + 3.77·16-s + (−16.7 − 16.7i)17-s + (−2.97 + 2.97i)18-s + 8.54i·19-s + ⋯
L(s)  = 1  + (−0.496 − 0.496i)2-s + (−0.408 + 0.408i)3-s − 0.507i·4-s + 0.405·6-s + (−0.267 − 0.267i)7-s + (−0.748 + 0.748i)8-s − 0.333i·9-s + 0.627·11-s + (0.206 + 0.206i)12-s + (0.914 − 0.914i)13-s + 0.265i·14-s + 0.235·16-s + (−0.988 − 0.988i)17-s + (−0.165 + 0.165i)18-s + 0.449i·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: $\chi_{525} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2255081196\)
\(L(\frac12)\) \(\approx\) \(0.2255081196\)
\(L(2)\) 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 \( 1 + (0.992 + 0.992i)T + 4iT^{2} \)
11 \( 1 - 6.89T + 121T^{2} \)
13 \( 1 + (-11.8 + 11.8i)T - 169iT^{2} \)
17 \( 1 + (16.7 + 16.7i)T + 289iT^{2} \)
19 \( 1 - 8.54iT - 361T^{2} \)
23 \( 1 + (12.4 - 12.4i)T - 529iT^{2} \)
29 \( 1 + 1.33iT - 841T^{2} \)
31 \( 1 + 18.4T + 961T^{2} \)
37 \( 1 + (31.4 + 31.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 26.7T + 1.68e3T^{2} \)
43 \( 1 + (-15.5 + 15.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (-22.1 - 22.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (66.4 - 66.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 81.8iT - 3.48e3T^{2} \)
61 \( 1 + 92.0T + 3.72e3T^{2} \)
67 \( 1 + (79.2 + 79.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 63.1T + 5.04e3T^{2} \)
73 \( 1 + (92.9 - 92.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 8.46iT - 6.24e3T^{2} \)
83 \( 1 + (36.2 - 36.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 32.5iT - 7.92e3T^{2} \)
97 \( 1 + (-79.2 - 79.2i)T + 9.40e3iT^{2} \)
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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

−10.30072801769436571821186484822, −9.306228751740557686212986882168, −8.800672119530509040767859794880, −7.43227943648608977620946253066, −6.22214478924805962462759043265, −5.56765898845495692430611802813, −4.30533715909751776131527974891, −3.07353258308385311023242960259, −1.46591582094718029936722527497, −0.11259425374299789889327555698, 1.79100151612219455850311929926, 3.44446627030490267122449238403, 4.50980681574205515933030108493, 6.25364765150956043012477142098, 6.47279257388609104440105780020, 7.51855462451980970239984940732, 8.701714212848875406965444653213, 8.925918792956080114252839618768, 10.22523471968506921060523039838, 11.32466201347192708353091229092

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