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Properties

Label 2-30e2-15.2-c3-0-6
Degree $2$
Conductor $900$
Sign $0.481 - 0.876i$
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  + (12.2 + 12.2i)7-s − 34.1i·11-s + (−25.8 + 25.8i)13-s + (15.5 − 15.5i)17-s + 32.1i·19-s + (72.9 + 72.9i)23-s + 106.·29-s + 10.3·31-s + (2.04 + 2.04i)37-s + 365. i·41-s + (10.2 − 10.2i)43-s + (260. − 260. i)47-s − 40.4i·49-s + (−42.4 − 42.4i)53-s − 375.·59-s + ⋯
L(s)  = 1  + (0.664 + 0.664i)7-s − 0.935i·11-s + (−0.550 + 0.550i)13-s + (0.222 − 0.222i)17-s + 0.387i·19-s + (0.661 + 0.661i)23-s + 0.681·29-s + 0.0601·31-s + (0.00907 + 0.00907i)37-s + 1.39i·41-s + (0.0363 − 0.0363i)43-s + (0.806 − 0.806i)47-s − 0.118i·49-s + (−0.109 − 0.109i)53-s − 0.828·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.481 - 0.876i$
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: \((2,\ 900,\ (\ :3/2),\ 0.481 - 0.876i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.994223533\)
\(L(\frac12)\) \(\approx\) \(1.994223533\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.2 - 12.2i)T + 343iT^{2} \)
11 \( 1 + 34.1iT - 1.33e3T^{2} \)
13 \( 1 + (25.8 - 25.8i)T - 2.19e3iT^{2} \)
17 \( 1 + (-15.5 + 15.5i)T - 4.91e3iT^{2} \)
19 \( 1 - 32.1iT - 6.85e3T^{2} \)
23 \( 1 + (-72.9 - 72.9i)T + 1.21e4iT^{2} \)
29 \( 1 - 106.T + 2.43e4T^{2} \)
31 \( 1 - 10.3T + 2.97e4T^{2} \)
37 \( 1 + (-2.04 - 2.04i)T + 5.06e4iT^{2} \)
41 \( 1 - 365. iT - 6.89e4T^{2} \)
43 \( 1 + (-10.2 + 10.2i)T - 7.95e4iT^{2} \)
47 \( 1 + (-260. + 260. i)T - 1.03e5iT^{2} \)
53 \( 1 + (42.4 + 42.4i)T + 1.48e5iT^{2} \)
59 \( 1 + 375.T + 2.05e5T^{2} \)
61 \( 1 + 770.T + 2.26e5T^{2} \)
67 \( 1 + (-603. - 603. i)T + 3.00e5iT^{2} \)
71 \( 1 - 695. iT - 3.57e5T^{2} \)
73 \( 1 + (262. - 262. i)T - 3.89e5iT^{2} \)
79 \( 1 + 32.2iT - 4.93e5T^{2} \)
83 \( 1 + (-519. - 519. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + (-416. - 416. i)T + 9.12e5iT^{2} \)
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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

−9.790087951178280736358888246353, −8.958044244309945328061391869333, −8.268264506485625391508872967597, −7.40855211183068370164733491427, −6.35424818916889631653312092755, −5.45688613005144696667163787468, −4.67202795326786077446007929681, −3.41005299515633862952529850873, −2.34166308868588483596746630501, −1.08433006102005756221149008189, 0.58794860492295579863199310932, 1.87317392995394512805811443867, 3.07543396685561441667611246435, 4.42127643843080743251558333428, 4.92824418264949571346951997613, 6.16005870357614831394697395434, 7.25182153933796430908476687004, 7.69927229999394775277560920745, 8.745764295709011175312748140150, 9.635990473737545466858591289460

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