Properties

Label 2-300-20.3-c1-0-17
Degree $2$
Conductor $300$
Sign $-0.793 + 0.608i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s + (0.707 − 0.707i)3-s + (−0.999 − 1.73i)4-s + (−0.366 − 1.36i)6-s + (−3.15 − 3.15i)7-s − 2.82·8-s − 1.00i·9-s + 2i·11-s + (−1.93 − 0.517i)12-s + (1.60 + 1.60i)13-s + (−6.09 + 1.63i)14-s + (−2.00 + 3.46i)16-s + (4.24 − 4.24i)17-s + (−1.22 − 0.707i)18-s + 3.19·19-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)2-s + (0.408 − 0.408i)3-s + (−0.499 − 0.866i)4-s + (−0.149 − 0.557i)6-s + (−1.19 − 1.19i)7-s − 0.999·8-s − 0.333i·9-s + 0.603i·11-s + (−0.557 − 0.149i)12-s + (0.444 + 0.444i)13-s + (−1.62 + 0.436i)14-s + (−0.500 + 0.866i)16-s + (1.02 − 1.02i)17-s + (−0.288 − 0.166i)18-s + 0.733·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.793 + 0.608i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.497811 - 1.46591i\)
\(L(\frac12)\) \(\approx\) \(0.497811 - 1.46591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (3.15 + 3.15i)T + 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (-1.60 - 1.60i)T + 13iT^{2} \)
17 \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \)
19 \( 1 - 3.19T + 19T^{2} \)
23 \( 1 + (-5.27 + 5.27i)T - 23iT^{2} \)
29 \( 1 + 0.535iT - 29T^{2} \)
31 \( 1 - 3.73iT - 31T^{2} \)
37 \( 1 + (7.72 - 7.72i)T - 37iT^{2} \)
41 \( 1 - 1.46T + 41T^{2} \)
43 \( 1 + (-3.15 + 3.15i)T - 43iT^{2} \)
47 \( 1 + (-0.656 - 0.656i)T + 47iT^{2} \)
53 \( 1 + (3.86 + 3.86i)T + 53iT^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + (-3.91 - 3.91i)T + 67iT^{2} \)
71 \( 1 + 2.53iT - 71T^{2} \)
73 \( 1 + (2.82 + 2.82i)T + 73iT^{2} \)
79 \( 1 + 7.46T + 79T^{2} \)
83 \( 1 + (9.14 - 9.14i)T - 83iT^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + (6.88 - 6.88i)T - 97iT^{2} \)
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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

−11.51785969909847222349176860495, −10.32004601403554989430147541871, −9.789447198223521327479445457610, −8.812482297429602015929136681637, −7.24010876229518134843330445776, −6.56801204793321543191218696998, −5.03934894693973852684617323264, −3.75480238412235295769257228767, −2.88190481429644464415862948045, −1.01698656125649746813744628889, 3.00907501542610901833139583488, 3.67359156283104065706766136618, 5.49767125447074587246551640626, 5.87324627083812429427244403403, 7.23220114149906870077315809172, 8.364513916457742784247977181971, 9.090105101431362284264469936339, 9.900287459264568523441845715594, 11.34044378987022737217213791451, 12.48968337807201693867402687391

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