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Properties

Label 2-300-60.47-c2-0-19
Degree $2$
Conductor $300$
Sign $-0.856 + 0.516i$
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  + (1.07 + 1.68i)2-s + (−0.130 + 2.99i)3-s + (−1.67 + 3.63i)4-s + (−5.18 + 3.01i)6-s + (1.91 − 1.91i)7-s + (−7.92 + 1.11i)8-s + (−8.96 − 0.782i)9-s − 6.87·11-s + (−10.6 − 5.47i)12-s + (−12.2 + 12.2i)13-s + (5.29 + 1.15i)14-s + (−10.4 − 12.1i)16-s + (−9.47 + 9.47i)17-s + (−8.36 − 15.9i)18-s + 33.2·19-s + ⋯
L(s)  = 1  + (0.539 + 0.841i)2-s + (−0.0434 + 0.999i)3-s + (−0.417 + 0.908i)4-s + (−0.864 + 0.502i)6-s + (0.273 − 0.273i)7-s + (−0.990 + 0.138i)8-s + (−0.996 − 0.0869i)9-s − 0.624·11-s + (−0.889 − 0.456i)12-s + (−0.944 + 0.944i)13-s + (0.378 + 0.0826i)14-s + (−0.651 − 0.758i)16-s + (−0.557 + 0.557i)17-s + (−0.464 − 0.885i)18-s + 1.75·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.856 + 0.516i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.856 + 0.516i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.359888 - 1.29278i\)
\(L(\frac12)\) \(\approx\) \(0.359888 - 1.29278i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 - 1.68i)T \)
3 \( 1 + (0.130 - 2.99i)T \)
5 \( 1 \)
good7 \( 1 + (-1.91 + 1.91i)T - 49iT^{2} \)
11 \( 1 + 6.87T + 121T^{2} \)
13 \( 1 + (12.2 - 12.2i)T - 169iT^{2} \)
17 \( 1 + (9.47 - 9.47i)T - 289iT^{2} \)
19 \( 1 - 33.2T + 361T^{2} \)
23 \( 1 + (7.20 - 7.20i)T - 529iT^{2} \)
29 \( 1 - 2.29T + 841T^{2} \)
31 \( 1 + 12.1iT - 961T^{2} \)
37 \( 1 + (-20.7 - 20.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 50.9iT - 1.68e3T^{2} \)
43 \( 1 + (15.1 + 15.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-26.7 - 26.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (15.5 + 15.5i)T + 2.80e3iT^{2} \)
59 \( 1 - 63.0iT - 3.48e3T^{2} \)
61 \( 1 - 28.4T + 3.72e3T^{2} \)
67 \( 1 + (32.4 - 32.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 88.8T + 5.04e3T^{2} \)
73 \( 1 + (71.1 - 71.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 75.1T + 6.24e3T^{2} \)
83 \( 1 + (-58.6 + 58.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 41.1T + 7.92e3T^{2} \)
97 \( 1 + (30.3 + 30.3i)T + 9.40e3iT^{2} \)
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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

−11.96217408489354955837825170830, −11.31721216695013974834343910616, −9.980770961145829052944985991221, −9.257925655225165818185400074751, −8.132881589680733414952062110222, −7.23378248310639523999917464176, −5.93862914745324270657046312519, −4.94732645388700577872399963795, −4.18383510718387161612361448832, −2.86084499573351793061904289027, 0.53186320617916996955278511228, 2.18912726475493493540680081813, 3.14086769393743189300236478365, 5.01649024130923259837350928273, 5.64450859244029661476530128392, 7.05009114854159618622535218335, 8.025784981766803212015952954655, 9.211121747527123237446793786286, 10.24870221044218634394515047301, 11.26181303612817571327607184171

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