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Properties

Label 2-300-60.23-c2-0-31
Degree $2$
Conductor $300$
Sign $-0.270 - 0.962i$
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  + (1.72 + 1.01i)2-s + (2.12 + 2.12i)3-s + (1.93 + 3.5i)4-s + (1.49 + 5.80i)6-s + (−0.218 + 7.99i)8-s + 8.99i·9-s + (−3.31 + 11.5i)12-s + (−8.50 + 13.5i)16-s + (−21.9 − 21.9i)17-s + (−9.14 + 15.5i)18-s + 30.9·19-s + (24.0 + 24.0i)23-s + (−17.4 + 16.4i)24-s + (−19.0 + 19.0i)27-s − 61.9i·31-s + (−28.4 + 14.7i)32-s + ⋯
L(s)  = 1  + (0.861 + 0.507i)2-s + (0.707 + 0.707i)3-s + (0.484 + 0.875i)4-s + (0.249 + 0.968i)6-s + (−0.0273 + 0.999i)8-s + 0.999i·9-s + (−0.276 + 0.961i)12-s + (−0.531 + 0.847i)16-s + (−1.28 − 1.28i)17-s + (−0.507 + 0.861i)18-s + 1.63·19-s + (1.04 + 1.04i)23-s + (−0.726 + 0.687i)24-s + (−0.707 + 0.707i)27-s − 1.99i·31-s + (−0.887 + 0.460i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.270 - 0.962i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.270 - 0.962i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.97315 + 2.60294i\)
\(L(\frac12)\) \(\approx\) \(1.97315 + 2.60294i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.72 - 1.01i)T \)
3 \( 1 + (-2.12 - 2.12i)T \)
5 \( 1 \)
good7 \( 1 + 49iT^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 + 169iT^{2} \)
17 \( 1 + (21.9 + 21.9i)T + 289iT^{2} \)
19 \( 1 - 30.9T + 361T^{2} \)
23 \( 1 + (-24.0 - 24.0i)T + 529iT^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + 61.9iT - 961T^{2} \)
37 \( 1 - 1.36e3iT^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 1.84e3iT^{2} \)
47 \( 1 + (-9.89 + 9.89i)T - 2.20e3iT^{2} \)
53 \( 1 + (43.8 - 43.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 118T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3iT^{2} \)
71 \( 1 + 5.04e3T^{2} \)
73 \( 1 + 5.32e3iT^{2} \)
79 \( 1 - 123.T + 6.24e3T^{2} \)
83 \( 1 + (108. + 108. i)T + 6.88e3iT^{2} \)
89 \( 1 + 7.92e3T^{2} \)
97 \( 1 - 9.40e3iT^{2} \)
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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

−11.65005839737493104944017053047, −11.18669897087237954950962369004, −9.691857863059472312752989452842, −8.973764813901972482602593827718, −7.76587586857408617955722773946, −7.04562121260620613048769206210, −5.51983364792617330956931408225, −4.69582940329239365535334113111, −3.53993689354968975616312395183, −2.48488419577926880032402090163, 1.27968170410496016501764370162, 2.62807049693864140519501641514, 3.69877064546487989777035772545, 5.02357581666162163697548968083, 6.38328294997343105751809164643, 7.08901677093439197321272917197, 8.432405133813837183744861221890, 9.367015422128321442882781290963, 10.52509107800629348279789919659, 11.42559080405930819774677683622

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