Properties

Label 2-300-4.3-c2-0-13
Degree $2$
Conductor $300$
Sign $0.176 - 0.984i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.177 + 1.99i)2-s − 1.73i·3-s + (−3.93 − 0.707i)4-s + (3.45 + 0.307i)6-s + 1.19i·7-s + (2.10 − 7.71i)8-s − 2.99·9-s + 8.22i·11-s + (−1.22 + 6.81i)12-s + 11.1·13-s + (−2.38 − 0.212i)14-s + (14.9 + 5.57i)16-s + 20.9·17-s + (0.533 − 5.97i)18-s + 27.9i·19-s + ⋯
L(s)  = 1  + (−0.0888 + 0.996i)2-s − 0.577i·3-s + (−0.984 − 0.176i)4-s + (0.575 + 0.0512i)6-s + 0.170i·7-s + (0.263 − 0.964i)8-s − 0.333·9-s + 0.747i·11-s + (−0.102 + 0.568i)12-s + 0.860·13-s + (−0.170 − 0.0151i)14-s + (0.937 + 0.348i)16-s + 1.23·17-s + (0.0296 − 0.332i)18-s + 1.47i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.176 - 0.984i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.176 - 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.06777 + 0.892895i\)
\(L(\frac12)\) \(\approx\) \(1.06777 + 0.892895i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.177 - 1.99i)T \)
3 \( 1 + 1.73iT \)
5 \( 1 \)
good7 \( 1 - 1.19iT - 49T^{2} \)
11 \( 1 - 8.22iT - 121T^{2} \)
13 \( 1 - 11.1T + 169T^{2} \)
17 \( 1 - 20.9T + 289T^{2} \)
19 \( 1 - 27.9iT - 361T^{2} \)
23 \( 1 + 9.48iT - 529T^{2} \)
29 \( 1 - 40.4T + 841T^{2} \)
31 \( 1 - 55.3iT - 961T^{2} \)
37 \( 1 - 50.1T + 1.36e3T^{2} \)
41 \( 1 + 73.6T + 1.68e3T^{2} \)
43 \( 1 + 19.0iT - 1.84e3T^{2} \)
47 \( 1 - 18.0iT - 2.20e3T^{2} \)
53 \( 1 + 57.2T + 2.80e3T^{2} \)
59 \( 1 + 60.6iT - 3.48e3T^{2} \)
61 \( 1 + 21.3T + 3.72e3T^{2} \)
67 \( 1 - 9.68iT - 4.48e3T^{2} \)
71 \( 1 + 68.6iT - 5.04e3T^{2} \)
73 \( 1 - 84.7T + 5.32e3T^{2} \)
79 \( 1 - 23.2iT - 6.24e3T^{2} \)
83 \( 1 - 93.2iT - 6.88e3T^{2} \)
89 \( 1 - 62.9T + 7.92e3T^{2} \)
97 \( 1 - 91.3T + 9.40e3T^{2} \)
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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

−12.17466324235716359283616423460, −10.54319059832735605349276521675, −9.732465556366801051816465398895, −8.496408211446446287942028589851, −7.920049646192105221826837243699, −6.81372520817343957494096154662, −6.00139273943919923642401969308, −4.91849961889336348036859847115, −3.47315332986363849169232329441, −1.32111593870796685139384114551, 0.859282072966255892317044452875, 2.80970547938812932508564302527, 3.80813336300188900308219360497, 4.94991419242877949969150510516, 6.09773145054618278981613160729, 7.83166289599705050398112387503, 8.729160849708474418967303412585, 9.604421618418289348526741806020, 10.45835345484521713573056863447, 11.29181673632783542713109119756

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