Properties

Label 2-30e2-9.5-c2-0-8
Degree $2$
Conductor $900$
Sign $0.213 - 0.976i$
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  + (−2.69 − 1.32i)3-s + (1.36 + 2.35i)7-s + (5.49 + 7.12i)9-s + (−2.77 + 1.60i)11-s + (3.99 − 6.92i)13-s − 1.82i·17-s + 15.4·19-s + (−0.544 − 8.14i)21-s + (−28.0 − 16.1i)23-s + (−5.37 − 26.4i)27-s + (−25.5 + 14.7i)29-s + (−26.5 + 45.9i)31-s + (9.58 − 0.641i)33-s − 30.0·37-s + (−19.9 + 13.3i)39-s + ⋯
L(s)  = 1  + (−0.897 − 0.441i)3-s + (0.194 + 0.336i)7-s + (0.610 + 0.791i)9-s + (−0.252 + 0.145i)11-s + (0.307 − 0.532i)13-s − 0.107i·17-s + 0.812·19-s + (−0.0259 − 0.387i)21-s + (−1.21 − 0.703i)23-s + (−0.199 − 0.979i)27-s + (−0.881 + 0.509i)29-s + (−0.854 + 1.48i)31-s + (0.290 − 0.0194i)33-s − 0.812·37-s + (−0.510 + 0.342i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.213 - 0.976i$
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: \((2,\ 900,\ (\ :1),\ 0.213 - 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8936885643\)
\(L(\frac12)\) \(\approx\) \(0.8936885643\)
\(L(2)\) 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 \( 1 + (2.69 + 1.32i)T \)
5 \( 1 \)
good7 \( 1 + (-1.36 - 2.35i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (2.77 - 1.60i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-3.99 + 6.92i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 1.82iT - 289T^{2} \)
19 \( 1 - 15.4T + 361T^{2} \)
23 \( 1 + (28.0 + 16.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (25.5 - 14.7i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (26.5 - 45.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 30.0T + 1.36e3T^{2} \)
41 \( 1 + (-66.5 - 38.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-4.49 - 7.77i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-10.2 + 5.92i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 28.9iT - 2.80e3T^{2} \)
59 \( 1 + (16.6 + 9.60i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-30.9 - 53.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-2.59 + 4.49i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 96.0iT - 5.04e3T^{2} \)
73 \( 1 - 127.T + 5.32e3T^{2} \)
79 \( 1 + (-24.2 - 41.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (83.1 - 48.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 21.4iT - 7.92e3T^{2} \)
97 \( 1 + (-83.3 - 144. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.32424743412669155079279762345, −9.314169619521025240722186957282, −8.250765785317439742313048543011, −7.49589955191026425837383317091, −6.63520242683854204577419172565, −5.63731949965753388704034627777, −5.11477493013279848867062591775, −3.84807439192139389516187630570, −2.40530138680136373156516416277, −1.14009025986448412396261503955, 0.37139647790393920562992864516, 1.86233965982165938019243894293, 3.63094129504658617413881745005, 4.29915330674576773006516139355, 5.50130795172802046449712726506, 6.01454900730586569637092016418, 7.18996552716357675151517297167, 7.86954464125893957694517177290, 9.216180249099745392524886366244, 9.714066984708673271442578748086

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