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Properties

Label 2-300-60.23-c2-0-40
Degree $2$
Conductor $300$
Sign $-0.582 + 0.812i$
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  + (−0.770 + 1.84i)2-s + (−2.78 − 1.12i)3-s + (−2.81 − 2.84i)4-s + (4.21 − 4.26i)6-s + (4.75 + 4.75i)7-s + (7.41 − 2.99i)8-s + (6.46 + 6.25i)9-s − 11.9·11-s + (4.61 + 11.0i)12-s + (4.22 + 4.22i)13-s + (−12.4 + 5.10i)14-s + (−0.188 + 15.9i)16-s + (−9.35 − 9.35i)17-s + (−16.5 + 7.11i)18-s − 1.48·19-s + ⋯
L(s)  = 1  + (−0.385 + 0.922i)2-s + (−0.927 − 0.375i)3-s + (−0.702 − 0.711i)4-s + (0.703 − 0.710i)6-s + (0.678 + 0.678i)7-s + (0.927 − 0.374i)8-s + (0.718 + 0.695i)9-s − 1.08·11-s + (0.384 + 0.922i)12-s + (0.324 + 0.324i)13-s + (−0.887 + 0.364i)14-s + (−0.0117 + 0.999i)16-s + (−0.550 − 0.550i)17-s + (−0.918 + 0.395i)18-s − 0.0783·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.582 + 0.812i$
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: \((2,\ 300,\ (\ :1),\ -0.582 + 0.812i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0152479 - 0.0296980i\)
\(L(\frac12)\) \(\approx\) \(0.0152479 - 0.0296980i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.770 - 1.84i)T \)
3 \( 1 + (2.78 + 1.12i)T \)
5 \( 1 \)
good7 \( 1 + (-4.75 - 4.75i)T + 49iT^{2} \)
11 \( 1 + 11.9T + 121T^{2} \)
13 \( 1 + (-4.22 - 4.22i)T + 169iT^{2} \)
17 \( 1 + (9.35 + 9.35i)T + 289iT^{2} \)
19 \( 1 + 1.48T + 361T^{2} \)
23 \( 1 + (11.6 + 11.6i)T + 529iT^{2} \)
29 \( 1 + 39.3T + 841T^{2} \)
31 \( 1 + 43.6iT - 961T^{2} \)
37 \( 1 + (49.1 - 49.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 27.0iT - 1.68e3T^{2} \)
43 \( 1 + (-8.84 + 8.84i)T - 1.84e3iT^{2} \)
47 \( 1 + (15.0 - 15.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (14.6 - 14.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 61.7iT - 3.48e3T^{2} \)
61 \( 1 + 84.6T + 3.72e3T^{2} \)
67 \( 1 + (65.7 + 65.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 14.2T + 5.04e3T^{2} \)
73 \( 1 + (16.0 + 16.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 9.32T + 6.24e3T^{2} \)
83 \( 1 + (12.7 + 12.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 52.4T + 7.92e3T^{2} \)
97 \( 1 + (6.90 - 6.90i)T - 9.40e3iT^{2} \)
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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

−11.10190180916359017545862812941, −10.31257716366854692760220416943, −9.130305028884476543797303159666, −8.094652614832457159794217288517, −7.32371731581680369932563816777, −6.18713710303142205716701565766, −5.38765903053907585728744371824, −4.52470968313612398604784623995, −1.92392397026183226437605375253, −0.02087130075185608190592517166, 1.63112082310822093813120087712, 3.53778555169381692129014026125, 4.61061438814899317403416724348, 5.57361061397272472533981877152, 7.19699374117808931372665687771, 8.130033035749169619414591811172, 9.287876796403279068342459455461, 10.49198370625920289174492656411, 10.70585726637611775481274059758, 11.55858577627130120900045958407

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