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Properties

Label 2-300-60.23-c2-0-19
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

\(L(\frac{3}{2})\) \(\approx\) \(0.661390 - 0.501363i\)
\(L(\frac12)\) \(\approx\) \(0.661390 - 0.501363i\)
\(L(2)\) 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

−11.42559142914584398215319975904, −10.35900992757346839348115788288, −9.720907629409369080712021969066, −8.225066600907801322997614144391, −7.71892159592698634086158201931, −6.56641736531206151023559500310, −5.54092696800952461417600111337, −3.79069753304414447664902870563, −2.17293845804058851813231785752, −0.796058186908413338193523867679, 1.02091244536737276253716445506, 3.27634854878612855406112279831, 5.07703734072734632689554588900, 5.66141602404010062268537310271, 6.94701364191740282857059974057, 7.80075909367924386337365016146, 9.163772260621936534618787300244, 9.773921373574092077001956093769, 10.51123742968189423589864640628, 11.65812686115643177983961293677

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