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Properties

Label 2-30e2-9.5-c2-0-2
Degree $2$
Conductor $900$
Sign $-0.0338 - 0.999i$
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  + (0.978 − 2.83i)3-s + (−2.71 − 4.69i)7-s + (−7.08 − 5.54i)9-s + (−8.81 + 5.09i)11-s + (−2.55 + 4.42i)13-s + 17.4i·17-s − 17.4·19-s + (−15.9 + 3.09i)21-s + (16.3 + 9.41i)23-s + (−22.6 + 14.6i)27-s + (29.0 − 16.7i)29-s + (−25.4 + 44.1i)31-s + (5.81 + 29.9i)33-s + 0.605·37-s + (10.0 + 11.5i)39-s + ⋯
L(s)  = 1  + (0.326 − 0.945i)3-s + (−0.387 − 0.671i)7-s + (−0.787 − 0.616i)9-s + (−0.801 + 0.462i)11-s + (−0.196 + 0.340i)13-s + 1.02i·17-s − 0.920·19-s + (−0.760 + 0.147i)21-s + (0.709 + 0.409i)23-s + (−0.839 + 0.543i)27-s + (1.00 − 0.578i)29-s + (−0.822 + 1.42i)31-s + (0.176 + 0.908i)33-s + 0.0163·37-s + (0.257 + 0.296i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0338 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0338 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.0338 - 0.999i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.0338 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4911128634\)
\(L(\frac12)\) \(\approx\) \(0.4911128634\)
\(L(2)\) 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 \( 1 + (-0.978 + 2.83i)T \)
5 \( 1 \)
good7 \( 1 + (2.71 + 4.69i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.81 - 5.09i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (2.55 - 4.42i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 17.4iT - 289T^{2} \)
19 \( 1 + 17.4T + 361T^{2} \)
23 \( 1 + (-16.3 - 9.41i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-29.0 + 16.7i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (25.4 - 44.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 0.605T + 1.36e3T^{2} \)
41 \( 1 + (12.4 + 7.17i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-25.4 - 44.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.96 - 1.13i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 18.4iT - 2.80e3T^{2} \)
59 \( 1 + (70.8 + 40.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-4.70 - 8.14i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-18.3 + 31.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 57.2iT - 5.04e3T^{2} \)
73 \( 1 + 78.9T + 5.32e3T^{2} \)
79 \( 1 + (19.1 + 33.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (66.9 - 38.6i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 138. iT - 7.92e3T^{2} \)
97 \( 1 + (-58.4 - 101. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.22451856876299202449626665486, −9.146703661104573167311958750923, −8.340636735547018562182648614816, −7.52864791405033778072204305178, −6.81865390893348788612472663949, −6.05432299594803442165191762099, −4.80903672760316572328226258947, −3.63800049832625423543087503567, −2.54135809407477212251473400925, −1.38460999975764634893715773759, 0.14858713454051849206540383368, 2.46663341084831395405931509492, 3.08088385583674740350140940252, 4.35193741734586864192545181736, 5.25620373847874608070846223913, 5.97998677774639873641199224979, 7.24355756851766937829556869107, 8.272315999152666945754993953812, 8.929076941122299655043002095557, 9.639890558813876354828335335887

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