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Properties

Label 2-300-60.23-c2-0-11
Degree $2$
Conductor $300$
Sign $0.672 - 0.740i$
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  + (−1.01 − 1.72i)2-s + (2.12 + 2.12i)3-s + (−1.93 + 3.5i)4-s + (1.49 − 5.80i)6-s + (7.99 − 0.218i)8-s + 8.99i·9-s + (−11.5 + 3.31i)12-s + (−8.50 − 13.5i)16-s + (21.9 + 21.9i)17-s + (15.5 − 9.14i)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 + (−14.7 + 28.4i)32-s + ⋯
L(s)  = 1  + (−0.507 − 0.861i)2-s + (0.707 + 0.707i)3-s + (−0.484 + 0.875i)4-s + (0.249 − 0.968i)6-s + (0.999 − 0.0273i)8-s + 0.999i·9-s + (−0.961 + 0.276i)12-s + (−0.531 − 0.847i)16-s + (1.28 + 1.28i)17-s + (0.861 − 0.507i)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 + (−0.460 + 0.887i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.672 - 0.740i$
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: \((2,\ 300,\ (\ :1),\ 0.672 - 0.740i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.26922 + 0.561944i\)
\(L(\frac12)\) \(\approx\) \(1.26922 + 0.561944i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.01 + 1.72i)T \)
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

−11.38803012846903332728831898804, −10.44701809200963881078651404739, −10.00044977449537024213490311435, −8.759130583837512130177859645239, −8.355157783353281298603771880943, −7.13405694795869541348087015109, −5.29491272905041150073550253557, −4.02716346567072023508153979067, −3.15922243004965806875682016909, −1.71599250869357659537540649365, 0.77795564418332026096952722812, 2.50540547546700592966998496545, 4.27935916635378175684146854658, 5.72601375040476373400947207411, 6.75185756304056668019540280083, 7.55795445344306814608959532147, 8.417072722374975392251281929333, 9.234454515926321144404301321385, 10.11579742883284440533631081416, 11.33565384478750009747360657376

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