Properties

Label 2-30e2-4.3-c2-0-52
Degree $2$
Conductor $900$
Sign $0.953 - 0.302i$
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  + (1.97 − 0.305i)2-s + (3.81 − 1.20i)4-s − 0.329i·7-s + (7.16 − 3.55i)8-s + 20.4i·11-s − 0.416·13-s + (−0.100 − 0.652i)14-s + (13.0 − 9.21i)16-s + 18.5·17-s + 12.4i·19-s + (6.26 + 40.5i)22-s + 23.2i·23-s + (−0.823 + 0.127i)26-s + (−0.398 − 1.25i)28-s + 23.9·29-s + ⋯
L(s)  = 1  + (0.988 − 0.152i)2-s + (0.953 − 0.302i)4-s − 0.0471i·7-s + (0.895 − 0.444i)8-s + 1.86i·11-s − 0.0320·13-s + (−0.00720 − 0.0465i)14-s + (0.817 − 0.575i)16-s + 1.09·17-s + 0.655i·19-s + (0.284 + 1.84i)22-s + 1.01i·23-s + (−0.0316 + 0.00489i)26-s + (−0.0142 − 0.0449i)28-s + 0.824·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.953 - 0.302i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.953 - 0.302i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.864386340\)
\(L(\frac12)\) \(\approx\) \(3.864386340\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.97 + 0.305i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 0.329iT - 49T^{2} \)
11 \( 1 - 20.4iT - 121T^{2} \)
13 \( 1 + 0.416T + 169T^{2} \)
17 \( 1 - 18.5T + 289T^{2} \)
19 \( 1 - 12.4iT - 361T^{2} \)
23 \( 1 - 23.2iT - 529T^{2} \)
29 \( 1 - 23.9T + 841T^{2} \)
31 \( 1 + 42.0iT - 961T^{2} \)
37 \( 1 - 50.9T + 1.36e3T^{2} \)
41 \( 1 + 46.7T + 1.68e3T^{2} \)
43 \( 1 + 55.5iT - 1.84e3T^{2} \)
47 \( 1 - 81.7iT - 2.20e3T^{2} \)
53 \( 1 - 29.9T + 2.80e3T^{2} \)
59 \( 1 + 24.3iT - 3.48e3T^{2} \)
61 \( 1 + 74.8T + 3.72e3T^{2} \)
67 \( 1 + 72.8iT - 4.48e3T^{2} \)
71 \( 1 + 39.2iT - 5.04e3T^{2} \)
73 \( 1 + 46.5T + 5.32e3T^{2} \)
79 \( 1 - 101. iT - 6.24e3T^{2} \)
83 \( 1 - 5.88iT - 6.88e3T^{2} \)
89 \( 1 - 61.0T + 7.92e3T^{2} \)
97 \( 1 - 95.5T + 9.40e3T^{2} \)
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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

−9.992498627348328364123756812729, −9.501459003012550557801461030674, −7.83195979473770757718663160864, −7.41067365671368153193033597961, −6.36114995807817704035662564816, −5.45862784064883387519483798348, −4.56540560249745524662933018344, −3.73564502301876154258415991809, −2.49857064745698613629257087570, −1.43075638842531843753231970581, 1.00967298909088852828550233605, 2.74085467949394956884131925052, 3.39985211422803931257754108365, 4.58024119202375945426950357719, 5.53431276605620151827281786959, 6.22679447727755905333590811922, 7.10063930905017140964016836578, 8.198523098671702365688039663608, 8.737752301658702545151466586092, 10.18343951108907758655050679799

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