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Properties

Label 2-450-5.3-c4-0-5
Degree $2$
Conductor $450$
Sign $0.229 - 0.973i$
Analytic cond. $46.5164$
Root an. cond. $6.82029$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2i)2-s − 8i·4-s + (19 − 19i)7-s + (−16 − 16i)8-s − 202·11-s + (99 + 99i)13-s − 76i·14-s − 64·16-s + (−239 + 239i)17-s − 40i·19-s + (−404 + 404i)22-s + (541 + 541i)23-s + 396·26-s + (−152 − 152i)28-s − 200i·29-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.5i·4-s + (0.387 − 0.387i)7-s + (−0.250 − 0.250i)8-s − 1.66·11-s + (0.585 + 0.585i)13-s − 0.387i·14-s − 0.250·16-s + (−0.826 + 0.826i)17-s − 0.110i·19-s + (−0.834 + 0.834i)22-s + (1.02 + 1.02i)23-s + 0.585·26-s + (−0.193 − 0.193i)28-s − 0.237i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(46.5164\)
Root analytic conductor: \(6.82029\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :2),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.177655738\)
\(L(\frac12)\) \(\approx\) \(1.177655738\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 2i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-19 + 19i)T - 2.40e3iT^{2} \)
11 \( 1 + 202T + 1.46e4T^{2} \)
13 \( 1 + (-99 - 99i)T + 2.85e4iT^{2} \)
17 \( 1 + (239 - 239i)T - 8.35e4iT^{2} \)
19 \( 1 + 40iT - 1.30e5T^{2} \)
23 \( 1 + (-541 - 541i)T + 2.79e5iT^{2} \)
29 \( 1 + 200iT - 7.07e5T^{2} \)
31 \( 1 + 758T + 9.23e5T^{2} \)
37 \( 1 + (141 - 141i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.04e3T + 2.82e6T^{2} \)
43 \( 1 + (-759 - 759i)T + 3.41e6iT^{2} \)
47 \( 1 + (459 - 459i)T - 4.87e6iT^{2} \)
53 \( 1 + (1.81e3 + 1.81e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 4.60e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.08e3T + 1.38e7T^{2} \)
67 \( 1 + (5.08e3 - 5.08e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 3.47e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.47e3 - 3.47e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 7.68e3iT - 3.89e7T^{2} \)
83 \( 1 + (-6.08e3 - 6.08e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 5.68e3iT - 6.27e7T^{2} \)
97 \( 1 + (561 - 561i)T - 8.85e7iT^{2} \)
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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

−10.96999253601733227639599294480, −10.01396631993002159022651932297, −8.922371224658770934622585117583, −7.937788245208023242520178803721, −6.92425026304366330855943559151, −5.72138955222983763643924306937, −4.83738942617845093871461634636, −3.80570161603564770186637160346, −2.57544236264127434810592184589, −1.37810788511410099850060729833, 0.25741579218555064151217511903, 2.26822528518633115307415035659, 3.28123628879066637036403378974, 4.82508294770116238484836714126, 5.32572834554432456518347281689, 6.49063078554468423026847434132, 7.53023156422885956153119614827, 8.313932747718202344688824671125, 9.145791811846195301443084102845, 10.54178138724810272434608563139

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