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Properties

Label 2-30e2-20.19-c2-0-40
Degree $2$
Conductor $900$
Sign $-0.116 - 0.993i$
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  + (1.75 + 0.950i)2-s + (2.19 + 3.34i)4-s + 3.98·7-s + (0.677 + 7.97i)8-s − 8.39i·11-s + 9.77i·13-s + (7.02 + 3.79i)14-s + (−6.38 + 14.6i)16-s + 12.1i·17-s + 17.3i·19-s + (7.97 − 14.7i)22-s + 2.97·23-s + (−9.28 + 17.1i)26-s + (8.74 + 13.3i)28-s + 51.6·29-s + ⋯
L(s)  = 1  + (0.879 + 0.475i)2-s + (0.548 + 0.836i)4-s + 0.569·7-s + (0.0847 + 0.996i)8-s − 0.763i·11-s + 0.751i·13-s + (0.501 + 0.270i)14-s + (−0.399 + 0.916i)16-s + 0.714i·17-s + 0.914i·19-s + (0.362 − 0.671i)22-s + 0.129·23-s + (−0.357 + 0.661i)26-s + (0.312 + 0.476i)28-s + 1.78·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.116 - 0.993i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.116 - 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.368618056\)
\(L(\frac12)\) \(\approx\) \(3.368618056\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.75 - 0.950i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 3.98T + 49T^{2} \)
11 \( 1 + 8.39iT - 121T^{2} \)
13 \( 1 - 9.77iT - 169T^{2} \)
17 \( 1 - 12.1iT - 289T^{2} \)
19 \( 1 - 17.3iT - 361T^{2} \)
23 \( 1 - 2.97T + 529T^{2} \)
29 \( 1 - 51.6T + 841T^{2} \)
31 \( 1 - 30.7iT - 961T^{2} \)
37 \( 1 + 19.5iT - 1.36e3T^{2} \)
41 \( 1 + 42.5T + 1.68e3T^{2} \)
43 \( 1 - 62.9T + 1.84e3T^{2} \)
47 \( 1 - 39.5T + 2.20e3T^{2} \)
53 \( 1 - 21.2iT - 2.80e3T^{2} \)
59 \( 1 - 50.3iT - 3.48e3T^{2} \)
61 \( 1 + 70.3T + 3.72e3T^{2} \)
67 \( 1 + 94.8T + 4.48e3T^{2} \)
71 \( 1 + 132. iT - 5.04e3T^{2} \)
73 \( 1 - 77.7iT - 5.32e3T^{2} \)
79 \( 1 - 48.3iT - 6.24e3T^{2} \)
83 \( 1 - 140.T + 6.88e3T^{2} \)
89 \( 1 + 18.1T + 7.92e3T^{2} \)
97 \( 1 - 15iT - 9.40e3T^{2} \)
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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

−10.44276647633480302969874701419, −8.975498737286263078756450472519, −8.333165852338436657855660092232, −7.53142245572817879592195888861, −6.50736760945198710406443285290, −5.83768283425967716194973369674, −4.81358141658484953411108737459, −3.98834700649244309104272735768, −2.92974223929323903386854474008, −1.58252086920188409067388665437, 0.845349793631593739470647691168, 2.23819298288090244412209212589, 3.14753424941637168440255625362, 4.51911726048614171660727775441, 4.95755458159361522051027197314, 6.06123848759861308340834047275, 7.01153617613607562583469150274, 7.82815263445442797099486989298, 9.022796015437569726365006861908, 9.941982272687975251374120256358

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