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Properties

Label 2-30e2-60.23-c0-0-1
Degree $2$
Conductor $900$
Sign $0.0618 - 0.998i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)8-s + (1 + i)13-s − 1.00·16-s + 1.41i·26-s − 1.41·29-s + (−0.707 − 0.707i)32-s + (1 − i)37-s − 1.41i·41-s − i·49-s + (−1.00 + 1.00i)52-s + (−1.00 − 1.00i)58-s − 1.00i·64-s + (−1 − i)73-s + 1.41·74-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)8-s + (1 + i)13-s − 1.00·16-s + 1.41i·26-s − 1.41·29-s + (−0.707 − 0.707i)32-s + (1 − i)37-s − 1.41i·41-s − i·49-s + (−1.00 + 1.00i)52-s + (−1.00 − 1.00i)58-s − 1.00i·64-s + (−1 − i)73-s + 1.41·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0618 - 0.998i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.0618 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.407472022\)
\(L(\frac12)\) \(\approx\) \(1.407472022\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−10.72901594796293752945233648299, −9.348543429743661986316643750505, −8.798227926560203511219294911946, −7.80259262735712497132045781850, −7.01262628319862680270419375074, −6.16134650254126951546869026065, −5.39153126851794812052790799203, −4.23569835035331771727100382637, −3.55404934073847338017082087436, −2.09895975816396534823669617728, 1.31908978340337804284569631485, 2.76999380682627280604358104954, 3.65827534333297483566749845420, 4.67514494859617523756613078518, 5.70653540675474887314163961979, 6.31993353994745031600812506942, 7.56266410809545128546414110557, 8.556043151348771228897729380235, 9.525916677704535156929471325783, 10.25608811938472251142208911767

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