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Properties

Label 2-30e2-15.2-c3-0-5
Degree $2$
Conductor $900$
Sign $0.998 - 0.0618i$
Analytic cond. $53.1017$
Root an. cond. $7.28709$
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  + (−17.9 − 17.9i)7-s − 4.24i·11-s + (−17.9 + 17.9i)13-s + (−76.0 + 76.0i)17-s + 26i·19-s + (76.0 + 76.0i)23-s + 110.·29-s + 52·31-s + (−17.9 − 17.9i)37-s − 199. i·41-s + (304. − 304. i)47-s + 299i·49-s + (−380. − 380. i)53-s + 717.·59-s + 350·61-s + ⋯
L(s)  = 1  + (−0.967 − 0.967i)7-s − 0.116i·11-s + (−0.382 + 0.382i)13-s + (−1.08 + 1.08i)17-s + 0.313i·19-s + (0.689 + 0.689i)23-s + 0.706·29-s + 0.301·31-s + (−0.0796 − 0.0796i)37-s − 0.759i·41-s + (0.943 − 0.943i)47-s + 0.871i·49-s + (−0.985 − 0.985i)53-s + 1.58·59-s + 0.734·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.998 - 0.0618i$
Analytic conductor: \(53.1017\)
Root analytic conductor: \(7.28709\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :3/2),\ 0.998 - 0.0618i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.434785961\)
\(L(\frac12)\) \(\approx\) \(1.434785961\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (17.9 + 17.9i)T + 343iT^{2} \)
11 \( 1 + 4.24iT - 1.33e3T^{2} \)
13 \( 1 + (17.9 - 17.9i)T - 2.19e3iT^{2} \)
17 \( 1 + (76.0 - 76.0i)T - 4.91e3iT^{2} \)
19 \( 1 - 26iT - 6.85e3T^{2} \)
23 \( 1 + (-76.0 - 76.0i)T + 1.21e4iT^{2} \)
29 \( 1 - 110.T + 2.43e4T^{2} \)
31 \( 1 - 52T + 2.97e4T^{2} \)
37 \( 1 + (17.9 + 17.9i)T + 5.06e4iT^{2} \)
41 \( 1 + 199. iT - 6.89e4T^{2} \)
43 \( 1 - 7.95e4iT^{2} \)
47 \( 1 + (-304. + 304. i)T - 1.03e5iT^{2} \)
53 \( 1 + (380. + 380. i)T + 1.48e5iT^{2} \)
59 \( 1 - 717.T + 2.05e5T^{2} \)
61 \( 1 - 350T + 2.26e5T^{2} \)
67 \( 1 + (-465. - 465. i)T + 3.00e5iT^{2} \)
71 \( 1 + 517. iT - 3.57e5T^{2} \)
73 \( 1 + (-465. + 465. i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.00e3iT - 4.93e5T^{2} \)
83 \( 1 + (-456. - 456. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.42e3T + 7.04e5T^{2} \)
97 \( 1 + (788. + 788. i)T + 9.12e5iT^{2} \)
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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

−9.821862665350421860923821705336, −8.956171153649641907236328352102, −8.056930845985154719993747718249, −6.93161397893554610771014420687, −6.57273731153437109507386420537, −5.36033515003413716502267116860, −4.17390057325959171944025558933, −3.48867648965415710000829148206, −2.14342019550228094336444655151, −0.68494914660814081721435389372, 0.58626881774160813572500315592, 2.43525952617514388908837400622, 2.99325641458106994421655714872, 4.46536808588274279722365361802, 5.31922522573587278775524853352, 6.38598767377500259185231482885, 6.96390276541700144596244958614, 8.132041426650352385034362008071, 9.078933817397964972802200107904, 9.509106071027473861845238644797

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