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Properties

Label 2-525-5.4-c3-0-38
Degree $2$
Conductor $525$
Sign $0.894 - 0.447i$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.52i·2-s + 3i·3-s + 5.66·4-s − 4.58·6-s − 7i·7-s + 20.8i·8-s − 9·9-s + 51.3·11-s + 16.9i·12-s − 87.2i·13-s + 10.6·14-s + 13.4·16-s − 80.6i·17-s − 13.7i·18-s + 29.8·19-s + ⋯
L(s)  = 1  + 0.540i·2-s + 0.577i·3-s + 0.708·4-s − 0.311·6-s − 0.377i·7-s + 0.922i·8-s − 0.333·9-s + 1.40·11-s + 0.408i·12-s − 1.86i·13-s + 0.204·14-s + 0.209·16-s − 1.15i·17-s − 0.180i·18-s + 0.360·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.643724810\)
\(L(\frac12)\) \(\approx\) \(2.643724810\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
7 \( 1 + 7iT \)
good2 \( 1 - 1.52iT - 8T^{2} \)
11 \( 1 - 51.3T + 1.33e3T^{2} \)
13 \( 1 + 87.2iT - 2.19e3T^{2} \)
17 \( 1 + 80.6iT - 4.91e3T^{2} \)
19 \( 1 - 29.8T + 6.85e3T^{2} \)
23 \( 1 - 1.71iT - 1.21e4T^{2} \)
29 \( 1 - 204.T + 2.43e4T^{2} \)
31 \( 1 + 150.T + 2.97e4T^{2} \)
37 \( 1 + 366. iT - 5.06e4T^{2} \)
41 \( 1 - 176.T + 6.89e4T^{2} \)
43 \( 1 - 394. iT - 7.95e4T^{2} \)
47 \( 1 + 507. iT - 1.03e5T^{2} \)
53 \( 1 + 149. iT - 1.48e5T^{2} \)
59 \( 1 + 463.T + 2.05e5T^{2} \)
61 \( 1 - 380.T + 2.26e5T^{2} \)
67 \( 1 - 797. iT - 3.00e5T^{2} \)
71 \( 1 + 220.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 - 111.T + 4.93e5T^{2} \)
83 \( 1 - 853. iT - 5.71e5T^{2} \)
89 \( 1 - 935.T + 7.04e5T^{2} \)
97 \( 1 + 783. iT - 9.12e5T^{2} \)
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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

−10.52605327857670187974417528094, −9.675364313526096151473652178203, −8.660200018069584384932653891710, −7.68613485012389521053672332851, −6.91448434800592244248953399997, −5.88664981400664940205224715327, −5.06706638153252031926342510534, −3.70524859212469475418206810106, −2.67492505287934107979698405325, −0.895247590825141523546003736999, 1.31431388383470481427002042107, 1.99765085623322345500252927533, 3.37407227219562581510372360650, 4.43465524003780882412488521306, 6.25435116632185342091351140990, 6.50078326185085895546509056741, 7.54026317264501867607345237419, 8.792987180366329031576708798140, 9.444507296471841078585963361868, 10.59184922942233939193739157267

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