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Properties

Label 2-525-5.4-c3-0-16
Degree $2$
Conductor $525$
Sign $-0.447 - 0.894i$
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  + 1.43i·2-s − 3i·3-s + 5.93·4-s + 4.31·6-s + 7i·7-s + 20.0i·8-s − 9·9-s + 7.61·11-s − 17.7i·12-s + 52.3i·13-s − 10.0·14-s + 18.6·16-s + 49.7i·17-s − 12.9i·18-s − 140.·19-s + ⋯
L(s)  = 1  + 0.508i·2-s − 0.577i·3-s + 0.741·4-s + 0.293·6-s + 0.377i·7-s + 0.885i·8-s − 0.333·9-s + 0.208·11-s − 0.428i·12-s + 1.11i·13-s − 0.192·14-s + 0.290·16-s + 0.709i·17-s − 0.169i·18-s − 1.69·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 525,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.756216714\)
\(L(\frac12)\) \(\approx\) \(1.756216714\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.43iT - 8T^{2} \)
11 \( 1 - 7.61T + 1.33e3T^{2} \)
13 \( 1 - 52.3iT - 2.19e3T^{2} \)
17 \( 1 - 49.7iT - 4.91e3T^{2} \)
19 \( 1 + 140.T + 6.85e3T^{2} \)
23 \( 1 + 23.4iT - 1.21e4T^{2} \)
29 \( 1 + 157.T + 2.43e4T^{2} \)
31 \( 1 - 127.T + 2.97e4T^{2} \)
37 \( 1 - 115. iT - 5.06e4T^{2} \)
41 \( 1 + 188.T + 6.89e4T^{2} \)
43 \( 1 - 322. iT - 7.95e4T^{2} \)
47 \( 1 - 76.6iT - 1.03e5T^{2} \)
53 \( 1 + 424. iT - 1.48e5T^{2} \)
59 \( 1 + 107.T + 2.05e5T^{2} \)
61 \( 1 - 915.T + 2.26e5T^{2} \)
67 \( 1 - 451. iT - 3.00e5T^{2} \)
71 \( 1 - 907.T + 3.57e5T^{2} \)
73 \( 1 - 755. iT - 3.89e5T^{2} \)
79 \( 1 + 22.5T + 4.93e5T^{2} \)
83 \( 1 - 1.11e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.51e3T + 7.04e5T^{2} \)
97 \( 1 - 549. iT - 9.12e5T^{2} \)
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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

−10.99364174527801513598750779898, −9.813164182012394865446654893605, −8.577770142033226400463424398554, −8.099929963366335391685513309538, −6.73876828635332411885197724590, −6.54189239517683060237137939159, −5.43872052990465124462062814153, −4.06709527517205054360542815519, −2.49264503942889908297833096289, −1.63780101468500349214939641569, 0.48899603228758433131339986425, 2.09159333960394245954018345206, 3.22456397247156522018546168291, 4.15923711529089998816759776223, 5.44616821086103789804405193978, 6.48265574308458684825144417685, 7.43023294125820936438976674549, 8.451108413335784301587286484962, 9.557175131080434770539843799139, 10.40675316361608626941193161411

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