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Properties

Label 2-300-15.14-c0-0-1
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

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

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