Properties

Label 2-300-15.14-c0-0-0
Degree $2$
Conductor $300$
Sign $0.447 - 0.894i$
Analytic cond. $0.149719$
Root an. cond. $0.386936$
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  + i·3-s + i·7-s − 9-s i·13-s + 19-s − 21-s i·27-s − 31-s − 2i·37-s + 39-s i·43-s + i·57-s − 61-s i·63-s + i·67-s + ⋯
L(s)  = 1  + i·3-s + i·7-s − 9-s i·13-s + 19-s − 21-s i·27-s − 31-s − 2i·37-s + 39-s i·43-s + i·57-s − 61-s i·63-s + i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(0.149719\)
Root analytic conductor: \(0.386936\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7589863419\)
\(L(\frac12)\) \(\approx\) \(0.7589863419\)
\(L(1)\) 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 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 - iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + 2iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−12.01372016912809342749285172174, −11.12320549034651232018049959416, −10.24535189912948783281184977550, −9.302592474543976489563525055459, −8.611554033917077913091286940942, −7.44423218924716347098249162256, −5.76908640453884171589561568387, −5.29423475529553211433386184685, −3.80587507684635087489723777263, −2.63549969015813420091347628242, 1.51565973822216203176734940741, 3.23349964907802010162953016143, 4.69009675734075972868552749914, 6.10915812908886137955795875640, 7.05929133007431612511485815057, 7.72608082543691412317778600171, 8.873022695828254400712013405300, 9.929993887286861233702550480290, 11.11616451565845491899036330357, 11.78061059495766263985688721443

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