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Properties

Label 2-1475-59.58-c0-0-5
Degree $2$
Conductor $1475$
Sign $-1$
Analytic cond. $0.736120$
Root an. cond. $0.857974$
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  − 1.41i·2-s − 1.00·4-s − 9-s − 1.41i·13-s − 0.999·16-s + 1.41i·18-s − 1.41i·23-s − 2.00·26-s + 1.41i·32-s + 1.00·36-s − 1.41i·37-s + 1.41i·43-s − 2.00·46-s − 1.41i·47-s − 49-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.00·4-s − 9-s − 1.41i·13-s − 0.999·16-s + 1.41i·18-s − 1.41i·23-s − 2.00·26-s + 1.41i·32-s + 1.00·36-s − 1.41i·37-s + 1.41i·43-s − 2.00·46-s − 1.41i·47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1475\)    =    \(5^{2} \cdot 59\)
Sign: $-1$
Analytic conductor: \(0.736120\)
Root analytic conductor: \(0.857974\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1475} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1475,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8608178823\)
\(L(\frac12)\) \(\approx\) \(0.8608178823\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
59 \( 1 - T \)
good2 \( 1 + 1.41iT - T^{2} \)
3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.41iT - T^{2} \)
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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

−9.526686414883737466004981706288, −8.648666571366913225727854546299, −8.019136638773275010020655443501, −6.82782471194764958260720039451, −5.83692501207181423016422091503, −4.94482706029995125024908061788, −3.82308343416056008318829456303, −2.96782634813814768949175756229, −2.26594889936624647880784999183, −0.67806292399861975845774580866, 1.98475609478405311410719508406, 3.35823307088419587551206029338, 4.57241829855701045579646929327, 5.34694528797564488199459493844, 6.17420791319497562684711211992, 6.77603651759951632737527942140, 7.63104556222187187241859609647, 8.337700370332096547381535784099, 9.079562784043829733195235290261, 9.661824760670272744223403996093

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