Properties

Label 2-450-5.2-c2-0-1
Degree $2$
Conductor $450$
Sign $-0.973 + 0.229i$
Analytic cond. $12.2616$
Root an. cond. $3.50165$
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 + i)2-s + 2i·4-s + (−3 − 3i)7-s + (−2 + 2i)8-s − 12·11-s + (−12 + 12i)13-s − 6i·14-s − 4·16-s + (−12 − 12i)17-s + 20i·19-s + (−12 − 12i)22-s + (−3 + 3i)23-s − 24·26-s + (6 − 6i)28-s + 30i·29-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (−0.428 − 0.428i)7-s + (−0.250 + 0.250i)8-s − 1.09·11-s + (−0.923 + 0.923i)13-s − 0.428i·14-s − 0.250·16-s + (−0.705 − 0.705i)17-s + 1.05i·19-s + (−0.545 − 0.545i)22-s + (−0.130 + 0.130i)23-s − 0.923·26-s + (0.214 − 0.214i)28-s + 1.03i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(12.2616\)
Root analytic conductor: \(3.50165\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5043332159\)
\(L(\frac12)\) \(\approx\) \(0.5043332159\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (3 + 3i)T + 49iT^{2} \)
11 \( 1 + 12T + 121T^{2} \)
13 \( 1 + (12 - 12i)T - 169iT^{2} \)
17 \( 1 + (12 + 12i)T + 289iT^{2} \)
19 \( 1 - 20iT - 361T^{2} \)
23 \( 1 + (3 - 3i)T - 529iT^{2} \)
29 \( 1 - 30iT - 841T^{2} \)
31 \( 1 + 8T + 961T^{2} \)
37 \( 1 + (48 + 48i)T + 1.36e3iT^{2} \)
41 \( 1 - 48T + 1.68e3T^{2} \)
43 \( 1 + (27 - 27i)T - 1.84e3iT^{2} \)
47 \( 1 + (27 + 27i)T + 2.20e3iT^{2} \)
53 \( 1 + (-12 + 12i)T - 2.80e3iT^{2} \)
59 \( 1 - 60iT - 3.48e3T^{2} \)
61 \( 1 - 32T + 3.72e3T^{2} \)
67 \( 1 + (3 + 3i)T + 4.48e3iT^{2} \)
71 \( 1 - 48T + 5.04e3T^{2} \)
73 \( 1 + (12 - 12i)T - 5.32e3iT^{2} \)
79 \( 1 - 40iT - 6.24e3T^{2} \)
83 \( 1 + (93 - 93i)T - 6.88e3iT^{2} \)
89 \( 1 + 30iT - 7.92e3T^{2} \)
97 \( 1 + (-12 - 12i)T + 9.40e3iT^{2} \)
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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

−11.45978368462385252032605115576, −10.44045677273259063978480810578, −9.564319432652994993661748681734, −8.531020441407849239558909366870, −7.39623409825557126083905220068, −6.88165710361960955613939425367, −5.61385385054849307645092436142, −4.72762904327044007051422065485, −3.59399985154746245982779657036, −2.26501707871681033292559392503, 0.16075624130832180634033572950, 2.27355510390479788293446068657, 3.12150638235640145834482265512, 4.59503965952966961387836556509, 5.43532237680396697126432624571, 6.46147206179342856635772207486, 7.63620326212253654900255948705, 8.662657052204385408560293276179, 9.760960734492452382314169658146, 10.44294953053361651290031001357

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