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Properties

Label 2-30e2-5.3-c2-0-1
Degree $2$
Conductor $900$
Sign $-0.525 - 0.850i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
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  + (2.44 − 2.44i)7-s − 6·11-s + (−12.2 − 12.2i)13-s + (−14.6 + 14.6i)17-s + 10i·19-s + (29.3 + 29.3i)23-s + 48i·29-s − 26·31-s + (31.8 − 31.8i)37-s − 30·41-s + (−29.3 − 29.3i)43-s + (14.6 − 14.6i)47-s + 37i·49-s + (−14.6 − 14.6i)53-s + 78i·59-s + ⋯
L(s)  = 1  + (0.349 − 0.349i)7-s − 0.545·11-s + (−0.942 − 0.942i)13-s + (−0.864 + 0.864i)17-s + 0.526i·19-s + (1.27 + 1.27i)23-s + 1.65i·29-s − 0.838·31-s + (0.860 − 0.860i)37-s − 0.731·41-s + (−0.683 − 0.683i)43-s + (0.312 − 0.312i)47-s + 0.755i·49-s + (−0.277 − 0.277i)53-s + 1.32i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7769063526\)
\(L(\frac12)\) \(\approx\) \(0.7769063526\)
\(L(2)\) 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 \( 1 + (-2.44 + 2.44i)T - 49iT^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 + (12.2 + 12.2i)T + 169iT^{2} \)
17 \( 1 + (14.6 - 14.6i)T - 289iT^{2} \)
19 \( 1 - 10iT - 361T^{2} \)
23 \( 1 + (-29.3 - 29.3i)T + 529iT^{2} \)
29 \( 1 - 48iT - 841T^{2} \)
31 \( 1 + 26T + 961T^{2} \)
37 \( 1 + (-31.8 + 31.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 30T + 1.68e3T^{2} \)
43 \( 1 + (29.3 + 29.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (-14.6 + 14.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (14.6 + 14.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 78iT - 3.48e3T^{2} \)
61 \( 1 - 2T + 3.72e3T^{2} \)
67 \( 1 + (63.6 - 63.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 120T + 5.04e3T^{2} \)
73 \( 1 + (-83.2 - 83.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 74iT - 6.24e3T^{2} \)
83 \( 1 + (44.0 + 44.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 150iT - 7.92e3T^{2} \)
97 \( 1 + (-4.89 + 4.89i)T - 9.40e3iT^{2} \)
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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

−10.40856258351209015559569173973, −9.335800070142543664538041629058, −8.507664122908614123664353332315, −7.56147285994215907222735921397, −7.02178768411567765806345527940, −5.65015929070116312502542301791, −5.04620310732192368520969275723, −3.85988694484407953008263160258, −2.77191517678768151072186394101, −1.43104362738419503809396537452, 0.24540925716943471800413213010, 2.07333524329183594228778075136, 2.90357349290286583319614447958, 4.59117989420214623872995806565, 4.90278136500489287606369625058, 6.28749672020372424188099641581, 7.04850385462747315471939168706, 7.932114384969730259542216627025, 8.916669317062875163912503294219, 9.496634540232571698719827058875

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