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Properties

Label 2-300-25.11-c1-0-3
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: $\chi_{300} (61, \cdot )$
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

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

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