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Properties

Label 2-30e2-45.29-c2-0-20
Degree $2$
Conductor $900$
Sign $0.974 - 0.225i$
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  + (2.92 − 0.686i)3-s + (7.89 + 4.55i)7-s + (8.05 − 4.00i)9-s + (−0.383 − 0.221i)11-s + (−9.62 + 5.55i)13-s + 8.01·17-s + 8.11·19-s + (26.1 + 7.89i)21-s + (11.8 + 20.4i)23-s + (20.7 − 17.2i)27-s + (45.9 + 26.5i)29-s + (−14.6 − 25.4i)31-s + (−1.27 − 0.383i)33-s − 18.4i·37-s + (−24.3 + 22.8i)39-s + ⋯
L(s)  = 1  + (0.973 − 0.228i)3-s + (1.12 + 0.651i)7-s + (0.895 − 0.445i)9-s + (−0.0348 − 0.0201i)11-s + (−0.740 + 0.427i)13-s + 0.471·17-s + 0.427·19-s + (1.24 + 0.375i)21-s + (0.513 + 0.888i)23-s + (0.769 − 0.638i)27-s + (1.58 + 0.913i)29-s + (−0.473 − 0.819i)31-s + (−0.0385 − 0.0116i)33-s − 0.499i·37-s + (−0.623 + 0.585i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.974 - 0.225i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.974 - 0.225i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.183412853\)
\(L(\frac12)\) \(\approx\) \(3.183412853\)
\(L(2)\) 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 \( 1 + (-2.92 + 0.686i)T \)
5 \( 1 \)
good7 \( 1 + (-7.89 - 4.55i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (0.383 + 0.221i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (9.62 - 5.55i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 8.01T + 289T^{2} \)
19 \( 1 - 8.11T + 361T^{2} \)
23 \( 1 + (-11.8 - 20.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-45.9 - 26.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.6 + 25.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 18.4iT - 1.36e3T^{2} \)
41 \( 1 + (38.9 - 22.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (19.9 + 11.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (4.22 - 7.32i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 60.5T + 2.80e3T^{2} \)
59 \( 1 + (65.9 - 38.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (2.67 - 4.63i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-95.0 + 54.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 16.0iT - 5.04e3T^{2} \)
73 \( 1 + 4.35iT - 5.32e3T^{2} \)
79 \( 1 + (-0.792 + 1.37i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-4.22 + 7.32i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 64.1iT - 7.92e3T^{2} \)
97 \( 1 + (-99.7 - 57.6i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.717773593707293861631734659524, −9.036532173468009384256434071571, −8.247152523339012640919252892520, −7.58735091868731312410625582161, −6.75354839906616844581641255926, −5.39559064618485004969775587512, −4.63438745626744088504095147507, −3.38325245664804263418409264853, −2.33601623303902940849505524529, −1.35035556493819475310231751458, 1.09382206351153583528933093801, 2.36743091357589865182625146099, 3.42598857852000877229716246306, 4.58043777452894818999644020998, 5.11636884296215765433892739217, 6.72298265927184917150278802045, 7.56771853982003988284005700019, 8.182414190562152869950784174676, 8.886038810803285157417655786260, 10.11061172526488618710199074007

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