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Properties

Label 2-30e2-5.2-c0-0-0
Degree $2$
Conductor $900$
Sign $0.793 - 0.608i$
Analytic cond. $0.449158$
Root an. cond. $0.670192$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)7-s + (−1.22 + 1.22i)13-s − i·19-s + 31-s + (1.22 − 1.22i)43-s + 1.99i·49-s − 61-s + (−1.22 − 1.22i)67-s − 2i·79-s − 2.99·91-s + (−1.22 − 1.22i)97-s + i·109-s + ⋯
L(s)  = 1  + (1.22 + 1.22i)7-s + (−1.22 + 1.22i)13-s − i·19-s + 31-s + (1.22 − 1.22i)43-s + 1.99i·49-s − 61-s + (−1.22 − 1.22i)67-s − 2i·79-s − 2.99·91-s + (−1.22 − 1.22i)97-s + i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.793 - 0.608i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.793 - 0.608i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.073134382\)
\(L(\frac12)\) \(\approx\) \(1.073134382\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
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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

−10.45650082203050754587648662324, −9.274047453835313919098768391461, −8.916301058786458780700295304279, −7.894100328866257288148004499645, −7.08475878432985113122910465677, −6.01040190620275579937618934991, −4.97757156259740452738938096395, −4.45645791053959658840702590127, −2.70906832935535461877131845257, −1.88259061974468479502304194211, 1.23196243106974681978028171987, 2.72978651449678672961196409348, 4.07139005917328997357403789229, 4.83848929404915519378501963603, 5.76283813340036200708337246177, 7.04773682711375936708027155857, 7.81775993782309514271072704526, 8.194432533002627210576222067735, 9.592621298271521156457216770300, 10.33800864541174707659782572476

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