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Properties

Label 2-300-25.16-c1-0-0
Degree $2$
Conductor $300$
Sign $-0.944 - 0.328i$
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  + (0.309 + 0.951i)3-s + (−1.49 + 1.66i)5-s − 4.78·7-s + (−0.809 + 0.587i)9-s + (−1.58 − 1.14i)11-s + (−0.873 + 0.634i)13-s + (−2.04 − 0.909i)15-s + (−1.17 + 3.61i)17-s + (1.31 − 4.04i)19-s + (−1.47 − 4.54i)21-s + (4.74 + 3.44i)23-s + (−0.522 − 4.97i)25-s + (−0.809 − 0.587i)27-s + (3.26 + 10.0i)29-s + (−1.33 + 4.10i)31-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (−0.669 + 0.743i)5-s − 1.80·7-s + (−0.269 + 0.195i)9-s + (−0.477 − 0.346i)11-s + (−0.242 + 0.176i)13-s + (−0.527 − 0.234i)15-s + (−0.285 + 0.877i)17-s + (0.301 − 0.927i)19-s + (−0.322 − 0.992i)21-s + (0.989 + 0.718i)23-s + (−0.104 − 0.994i)25-s + (−0.155 − 0.113i)27-s + (0.605 + 1.86i)29-s + (−0.239 + 0.737i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.944 - 0.328i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.944 - 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0877302 + 0.518692i\)
\(L(\frac12)\) \(\approx\) \(0.0877302 + 0.518692i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (1.49 - 1.66i)T \)
good7 \( 1 + 4.78T + 7T^{2} \)
11 \( 1 + (1.58 + 1.14i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.873 - 0.634i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.17 - 3.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.31 + 4.04i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-4.74 - 3.44i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-3.26 - 10.0i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.33 - 4.10i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.57 - 3.32i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.694 - 0.504i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (-0.927 - 2.85i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.30 - 4.01i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.85 + 2.80i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.93 + 2.13i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.14 + 6.59i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.70 + 11.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-13.7 - 9.96i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.04 + 6.29i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.797 - 2.45i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-0.673 - 0.489i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.81 + 8.67i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.17909928545689529440090578852, −10.96409919077239009178156278455, −10.38327248840522999012006097078, −9.404777793254738392109839011354, −8.524366915266454417040662452888, −7.11028056441972544991683686747, −6.50832075327443187932744727582, −5.05498044443543253267349679617, −3.48591427016371792279880665852, −3.01848874051742557511064495197, 0.35959520115248816312758783364, 2.67694257915881551962844570464, 3.87718474866200714560064479084, 5.32080992557715849671829713460, 6.55856735290498156648593683294, 7.40847736745945093360533869768, 8.434163247634725654815031199060, 9.446904002261978257724595169963, 10.16702317853417903133627195062, 11.65942838157852081597909747160

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