Properties

Label 2-300-300.191-c1-0-47
Degree $2$
Conductor $300$
Sign $0.781 + 0.623i$
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  + (1.40 + 0.124i)2-s + (−0.392 − 1.68i)3-s + (1.96 + 0.350i)4-s + (2.20 − 0.380i)5-s + (−0.343 − 2.42i)6-s + 0.0822i·7-s + (2.73 + 0.738i)8-s + (−2.69 + 1.32i)9-s + (3.15 − 0.262i)10-s + (−2.26 − 1.64i)11-s + (−0.182 − 3.45i)12-s + (−3.85 + 2.80i)13-s + (−0.0102 + 0.115i)14-s + (−1.50 − 3.56i)15-s + (3.75 + 1.38i)16-s + (3.83 + 1.24i)17-s + ⋯
L(s)  = 1  + (0.996 + 0.0879i)2-s + (−0.226 − 0.973i)3-s + (0.984 + 0.175i)4-s + (0.985 − 0.170i)5-s + (−0.140 − 0.990i)6-s + 0.0310i·7-s + (0.965 + 0.261i)8-s + (−0.897 + 0.441i)9-s + (0.996 − 0.0829i)10-s + (−0.681 − 0.495i)11-s + (−0.0525 − 0.998i)12-s + (−1.06 + 0.777i)13-s + (−0.00273 + 0.0309i)14-s + (−0.389 − 0.921i)15-s + (0.938 + 0.345i)16-s + (0.929 + 0.302i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.781 + 0.623i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.781 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22660 - 0.779372i\)
\(L(\frac12)\) \(\approx\) \(2.22660 - 0.779372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.124i)T \)
3 \( 1 + (0.392 + 1.68i)T \)
5 \( 1 + (-2.20 + 0.380i)T \)
good7 \( 1 - 0.0822iT - 7T^{2} \)
11 \( 1 + (2.26 + 1.64i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.85 - 2.80i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.83 - 1.24i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.03 + 0.661i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.93 + 1.40i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-3.40 + 1.10i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.91 + 0.947i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.33 - 5.32i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.77 + 6.57i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 - 8.76iT - 43T^{2} \)
47 \( 1 + (-0.650 - 2.00i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.15 - 2.32i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.56 - 5.49i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.43 + 1.77i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-12.2 - 3.97i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.11 - 3.42i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.23 + 0.898i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.51 + 1.46i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.96 + 9.13i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-6.21 + 8.55i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.56 + 7.89i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.09202060194362611932362727501, −10.92456895241619618100819327764, −10.02513478977972741103692968556, −8.536919167239741503307818250555, −7.47413027093643409315956909289, −6.51069249316697588841156488030, −5.70254387783375896228815535367, −4.81156139854972753445760185836, −2.88637888973724154554869550019, −1.81222085570209628760601115503, 2.35924924566440180895003520880, 3.49928529351613145809773175592, 5.07282128368124085829338520947, 5.36553039123121320785378466381, 6.61064085099658751829614817081, 7.83918352123552403230320159234, 9.445823687396431197744009294467, 10.30576245653894662747194283921, 10.63814710109599438447143998099, 12.04750346371552419581410044981

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