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Properties

Label 2-30e2-45.29-c2-0-28
Degree $2$
Conductor $900$
Sign $-0.118 + 0.992i$
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  + (0.783 − 2.89i)3-s + (−4.08 − 2.35i)7-s + (−7.77 − 4.53i)9-s + (16.6 + 9.63i)11-s + (8.11 − 4.68i)13-s + 16.2·17-s + 19.0·19-s + (−10.0 + 9.97i)21-s + (−8.30 − 14.3i)23-s + (−19.2 + 18.9i)27-s + (3.87 + 2.23i)29-s + (−7.19 − 12.4i)31-s + (40.9 − 40.7i)33-s − 55.6i·37-s + (−7.21 − 27.1i)39-s + ⋯
L(s)  = 1  + (0.261 − 0.965i)3-s + (−0.583 − 0.336i)7-s + (−0.863 − 0.504i)9-s + (1.51 + 0.875i)11-s + (0.624 − 0.360i)13-s + 0.955·17-s + 1.00·19-s + (−0.477 + 0.474i)21-s + (−0.361 − 0.625i)23-s + (−0.712 + 0.702i)27-s + (0.133 + 0.0771i)29-s + (−0.232 − 0.401i)31-s + (1.24 − 1.23i)33-s − 1.50i·37-s + (−0.184 − 0.696i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.118 + 0.992i$
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: \((2,\ 900,\ (\ :1),\ -0.118 + 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.052668872\)
\(L(\frac12)\) \(\approx\) \(2.052668872\)
\(L(2)\) 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 \( 1 + (-0.783 + 2.89i)T \)
5 \( 1 \)
good7 \( 1 + (4.08 + 2.35i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-16.6 - 9.63i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.11 + 4.68i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 16.2T + 289T^{2} \)
19 \( 1 - 19.0T + 361T^{2} \)
23 \( 1 + (8.30 + 14.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-3.87 - 2.23i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (7.19 + 12.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 55.6iT - 1.36e3T^{2} \)
41 \( 1 + (-26.7 + 15.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (37.3 + 21.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-20.7 + 35.9i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 30.7T + 2.80e3T^{2} \)
59 \( 1 + (74.4 - 42.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (51.7 - 89.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-107. + 61.7i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 111. iT - 5.04e3T^{2} \)
73 \( 1 - 90.5iT - 5.32e3T^{2} \)
79 \( 1 + (41.6 - 72.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-17.9 + 31.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 84.9iT - 7.92e3T^{2} \)
97 \( 1 + (-91.1 - 52.6i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.515932698900960148179846928344, −8.916618256919448585571972557848, −7.81566134273433084200060661747, −7.14230306093103431682522519918, −6.39959766298795855363713552549, −5.56294725172946010649997145694, −4.02374882749803221967989428322, −3.22631594023713765579230639996, −1.81723069141540274324482249531, −0.74113846212865695696769173202, 1.29847903932042816675000155743, 3.17455741355133793447684893059, 3.55524085083015667198671763696, 4.74460051917487726433843336871, 5.86581265391653657521585803556, 6.43471567204375157565063733737, 7.82748311054410692222260680189, 8.664359417298099728211177099680, 9.481001240351016962742159563978, 9.787211357004048254386601492545

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