Properties

Label 2-300-12.11-c1-0-16
Degree $2$
Conductor $300$
Sign $0.997 + 0.0642i$
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  + (0.273 + 1.38i)2-s + (−1.55 − 0.758i)3-s + (−1.85 + 0.758i)4-s + (0.626 − 2.36i)6-s − 3.56i·7-s + (−1.55 − 2.36i)8-s + (1.85 + 2.36i)9-s + 4.20·11-s + (3.45 + 0.222i)12-s + 2.70·13-s + (4.94 − 0.973i)14-s + (2.85 − 2.80i)16-s − 0.828i·17-s + (−2.77 + 3.21i)18-s − 5.07i·19-s + ⋯
L(s)  = 1  + (0.193 + 0.981i)2-s + (−0.899 − 0.437i)3-s + (−0.925 + 0.379i)4-s + (0.255 − 0.966i)6-s − 1.34i·7-s + (−0.550 − 0.834i)8-s + (0.616 + 0.787i)9-s + 1.26·11-s + (0.997 + 0.0642i)12-s + 0.749·13-s + (1.32 − 0.260i)14-s + (0.712 − 0.701i)16-s − 0.200i·17-s + (−0.653 + 0.757i)18-s − 1.16i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.997 + 0.0642i$
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: \((2,\ 300,\ (\ :1/2),\ 0.997 + 0.0642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.991339 - 0.0318579i\)
\(L(\frac12)\) \(\approx\) \(0.991339 - 0.0318579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.273 - 1.38i)T \)
3 \( 1 + (1.55 + 0.758i)T \)
5 \( 1 \)
good7 \( 1 + 3.56iT - 7T^{2} \)
11 \( 1 - 4.20T + 11T^{2} \)
13 \( 1 - 2.70T + 13T^{2} \)
17 \( 1 + 0.828iT - 17T^{2} \)
19 \( 1 + 5.07iT - 19T^{2} \)
23 \( 1 + 1.09T + 23T^{2} \)
29 \( 1 + 5.55iT - 29T^{2} \)
31 \( 1 - 6.59iT - 31T^{2} \)
37 \( 1 - 5.40T + 37T^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + 0.531iT - 43T^{2} \)
47 \( 1 + 6.22T + 47T^{2} \)
53 \( 1 - 5.55iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 0.701T + 61T^{2} \)
67 \( 1 - 2.04iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 7.70T + 73T^{2} \)
79 \( 1 - 7.12iT - 79T^{2} \)
83 \( 1 + 3.11T + 83T^{2} \)
89 \( 1 - 4.72iT - 89T^{2} \)
97 \( 1 + 8.10T + 97T^{2} \)
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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

−11.78598088701346967877957424199, −10.89864424648501642005511857166, −9.830067157257198648443103854657, −8.683744741153598643136829673568, −7.44945751213071356992752393114, −6.82653652487426727345357804153, −6.04474814931567982762350873513, −4.70914271532875139571447180037, −3.85252059322530232028444401677, −0.900346535699855662153296117377, 1.56862960486840238227799473619, 3.40876520709646186347334872651, 4.47181657406146890817701410400, 5.72893380692458335137116544502, 6.28184847063273008085826292240, 8.343767753871889142743952036898, 9.276319735951225835416826017218, 9.931137944822739982260508234982, 11.13106645817281781871223051581, 11.68029079216577329722913123703

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