Properties

Label 2-300-12.11-c1-0-12
Degree $2$
Conductor $300$
Sign $0.408 - 0.912i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (−0.707 + 1.58i)3-s + 2.00·4-s + (−1.00 + 2.23i)6-s + 3.16i·7-s + 2.82·8-s + (−2.00 − 2.23i)9-s + (−1.41 + 3.16i)12-s + 4.47i·14-s + 4.00·16-s + (−2.82 − 3.16i)18-s + (−5.00 − 2.23i)21-s + 1.41·23-s + (−2.00 + 4.47i)24-s + (4.94 − 1.58i)27-s + 6.32i·28-s + ⋯
L(s)  = 1  + 1.00·2-s + (−0.408 + 0.912i)3-s + 1.00·4-s + (−0.408 + 0.912i)6-s + 1.19i·7-s + 1.00·8-s + (−0.666 − 0.745i)9-s + (−0.408 + 0.912i)12-s + 1.19i·14-s + 1.00·16-s + (−0.666 − 0.745i)18-s + (−1.09 − 0.487i)21-s + 0.294·23-s + (−0.408 + 0.912i)24-s + (0.952 − 0.304i)27-s + 1.19i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.408 - 0.912i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73141 + 1.12235i\)
\(L(\frac12)\) \(\approx\) \(1.73141 + 1.12235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (0.707 - 1.58i)T \)
5 \( 1 \)
good7 \( 1 - 3.16iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 1.41T + 23T^{2} \)
29 \( 1 + 8.94iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 4.47iT - 41T^{2} \)
43 \( 1 + 3.16iT - 43T^{2} \)
47 \( 1 + 9.89T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 15.8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 - 17.8iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.86510524643463939715639775320, −11.26620435365702918562014884633, −10.22210358065822550779569645023, −9.253797081645866551219571016603, −8.099889564467530799192740993572, −6.55760537413229617345144475204, −5.70347239274477912542262981235, −4.90793919715533224343542719492, −3.72870891484257363571015925605, −2.46962120675756535733012006642, 1.40846540252742740862468046415, 3.09350347636820903833734666410, 4.48114490032672114408415440880, 5.56387658201116029539267847275, 6.74966095109870690982018776324, 7.25271579290364933410954135621, 8.323886409467531876953720789446, 10.11501064144395588854236741710, 10.99154064805850309928776110131, 11.64132590055451221544924785125

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