Properties

Label 2-300-15.2-c1-0-1
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: $\chi_{300} (257, \cdot )$
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

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

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