Properties

Label 2-300-15.8-c1-0-5
Degree $2$
Conductor $300$
Sign $0.187 + 0.982i$
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  + (0.618 − 1.61i)3-s + (1 − i)7-s + (−2.23 − 2.00i)9-s − 4.47i·11-s + (3 + 3i)13-s + (−2.23 − 2.23i)17-s + 2i·19-s + (−1 − 2.23i)21-s + (2.23 − 2.23i)23-s + (−4.61 + 2.38i)27-s − 4.47·29-s + 4·31-s + (−7.23 − 2.76i)33-s + (3 − 3i)37-s + (6.70 − 3i)39-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s + (0.377 − 0.377i)7-s + (−0.745 − 0.666i)9-s − 1.34i·11-s + (0.832 + 0.832i)13-s + (−0.542 − 0.542i)17-s + 0.458i·19-s + (−0.218 − 0.487i)21-s + (0.466 − 0.466i)23-s + (−0.888 + 0.458i)27-s − 0.830·29-s + 0.718·31-s + (−1.25 − 0.481i)33-s + (0.493 − 0.493i)37-s + (1.07 − 0.480i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.187 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11632 - 0.923303i\)
\(L(\frac12)\) \(\approx\) \(1.11632 - 0.923303i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.618 + 1.61i)T \)
5 \( 1 \)
good7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (2.23 + 2.23i)T + 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-2.23 + 2.23i)T - 23iT^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 8.94iT - 41T^{2} \)
43 \( 1 + (-3 - 3i)T + 43iT^{2} \)
47 \( 1 + (-6.70 - 6.70i)T + 47iT^{2} \)
53 \( 1 + (2.23 - 2.23i)T - 53iT^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-1 + i)T - 67iT^{2} \)
71 \( 1 - 4.47iT - 71T^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (6.70 - 6.70i)T - 83iT^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + (9 - 9i)T - 97iT^{2} \)
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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

−11.35391331091906233561861649643, −11.04353774483908957040979734154, −9.375542861506582443292378434625, −8.569981110444278696405843356746, −7.75805406001279014910180815952, −6.64238890294253049764031788464, −5.82979013310418854009129207376, −4.15978367529965748139362840761, −2.81455434271814746271916508568, −1.16587846828609069376831781271, 2.22398376156770181511611763688, 3.69176098081617736417805070568, 4.78111411192609383689351198856, 5.75087394928268454706433974125, 7.24355413676920567416071509382, 8.361699605012605427135800660692, 9.109105243283859781380181217322, 10.11426050881892271880340441366, 10.85166698912862104749893136814, 11.79393281739282728100547967585

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