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Properties

Label 2-1500-15.2-c1-0-9
Degree $2$
Conductor $1500$
Sign $-0.656 - 0.753i$
Analytic cond. $11.9775$
Root an. cond. $3.46086$
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  + (1.13 + 1.30i)3-s + (1.30 + 1.30i)7-s + (−0.410 + 2.97i)9-s + 4.73i·11-s + (−3.67 + 3.67i)13-s + (1.75 − 1.75i)17-s − 2.67i·19-s + (−0.218 + 3.17i)21-s + (−3.08 − 3.08i)23-s + (−4.34 + 2.84i)27-s + 6.06·29-s − 4.32·31-s + (−6.17 + 5.38i)33-s + (−5.07 − 5.07i)37-s + (−8.98 − 0.618i)39-s + ⋯
L(s)  = 1  + (0.656 + 0.753i)3-s + (0.491 + 0.491i)7-s + (−0.136 + 0.990i)9-s + 1.42i·11-s + (−1.01 + 1.01i)13-s + (0.425 − 0.425i)17-s − 0.613i·19-s + (−0.0477 + 0.693i)21-s + (−0.643 − 0.643i)23-s + (−0.836 + 0.547i)27-s + 1.12·29-s − 0.776·31-s + (−1.07 + 0.937i)33-s + (−0.834 − 0.834i)37-s + (−1.43 − 0.0989i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.656 - 0.753i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.656 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.822113947\)
\(L(\frac12)\) \(\approx\) \(1.822113947\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 - 1.30i)T \)
5 \( 1 \)
good7 \( 1 + (-1.30 - 1.30i)T + 7iT^{2} \)
11 \( 1 - 4.73iT - 11T^{2} \)
13 \( 1 + (3.67 - 3.67i)T - 13iT^{2} \)
17 \( 1 + (-1.75 + 1.75i)T - 17iT^{2} \)
19 \( 1 + 2.67iT - 19T^{2} \)
23 \( 1 + (3.08 + 3.08i)T + 23iT^{2} \)
29 \( 1 - 6.06T + 29T^{2} \)
31 \( 1 + 4.32T + 31T^{2} \)
37 \( 1 + (5.07 + 5.07i)T + 37iT^{2} \)
41 \( 1 - 2.12iT - 41T^{2} \)
43 \( 1 + (7.19 - 7.19i)T - 43iT^{2} \)
47 \( 1 + (-1.35 + 1.35i)T - 47iT^{2} \)
53 \( 1 + (-5.06 - 5.06i)T + 53iT^{2} \)
59 \( 1 - 0.851T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + (-5.53 - 5.53i)T + 67iT^{2} \)
71 \( 1 + 8.77iT - 71T^{2} \)
73 \( 1 + (8.35 - 8.35i)T - 73iT^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 + (-9.52 - 9.52i)T + 83iT^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + (1.08 + 1.08i)T + 97iT^{2} \)
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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

−9.777716300836692527728128166312, −9.057653356302573686843555309387, −8.320190548247889739692395000893, −7.40694834844926911196997318517, −6.75816726934171500862505401707, −5.25115999422289492570284802070, −4.76003781223208118500275618816, −3.97509490046785611067615818924, −2.58360803060422338001652826063, −1.97926390783779787226406280776, 0.63905265884333752129191847027, 1.85961139662514215841370685387, 3.11606189746415376669098437300, 3.73951971360016720611498800779, 5.17523365593584525097740029756, 5.94368631397968923589718982500, 6.91611673259053007828544627307, 7.81378947967670404227830205654, 8.179360431078740193749314336129, 8.956035691317859100930202575589

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