Properties

Label 2-300-12.11-c1-0-27
Degree $2$
Conductor $300$
Sign $0.250 + 0.968i$
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  + (0.866 − 1.11i)2-s + 1.73·3-s + (−0.500 − 1.93i)4-s + (1.49 − 1.93i)6-s + (−2.59 − 1.11i)8-s + 2.99·9-s + (−0.866 − 3.35i)12-s + (−3.5 + 1.93i)16-s − 4.47i·17-s + (2.59 − 3.35i)18-s + 7.74i·19-s − 3.46·23-s + (−4.50 − 1.93i)24-s + 5.19·27-s + 7.74i·31-s + (−0.866 + 5.59i)32-s + ⋯
L(s)  = 1  + (0.612 − 0.790i)2-s + 1.00·3-s + (−0.250 − 0.968i)4-s + (0.612 − 0.790i)6-s + (−0.918 − 0.395i)8-s + 0.999·9-s + (−0.250 − 0.968i)12-s + (−0.875 + 0.484i)16-s − 1.08i·17-s + (0.612 − 0.790i)18-s + 1.77i·19-s − 0.722·23-s + (−0.918 − 0.395i)24-s + 1.00·27-s + 1.39i·31-s + (−0.153 + 0.988i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.250 + 0.968i$
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: \((2,\ 300,\ (\ :1/2),\ 0.250 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78369 - 1.38164i\)
\(L(\frac12)\) \(\approx\) \(1.78369 - 1.38164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 1.11i)T \)
3 \( 1 - 1.73T \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 4.47iT - 17T^{2} \)
19 \( 1 - 7.74iT - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 7.74iT - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + 4.47iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
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

−11.78175675085698842844272736484, −10.46947671026963346654234935341, −9.843221995395319042573180136518, −8.901599818302637925059394638066, −7.83743197920996102622876190385, −6.56238621750688459368607035467, −5.21252859413873842000939297548, −4.00566856173883031188541835477, −3.03187788127953670040836800673, −1.69383047641562142343578541543, 2.45657223494028021055502165666, 3.74233123488470895105617752460, 4.70492407682606185113424540184, 6.12363996841608109918257343483, 7.14311312029625923470584536406, 8.042117752955831477681839279600, 8.834857352242505299862297906693, 9.753943823661843734144148892787, 11.11465149098388891745483785302, 12.29659070470661254217262546053

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