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Properties

Label 2-2400-20.7-c1-0-16
Degree $2$
Conductor $2400$
Sign $0.991 + 0.130i$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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 − 0.707i)3-s + (0.0249 − 0.0249i)7-s + 1.00i·9-s − 1.86i·11-s + (−0.189 + 0.189i)13-s + (4.14 + 4.14i)17-s − 4.13·19-s − 0.0352·21-s + (3.18 + 3.18i)23-s + (0.707 − 0.707i)27-s − 2.42i·29-s − 3.52i·31-s + (−1.31 + 1.31i)33-s + (2.63 + 2.63i)37-s + 0.267·39-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.00942 − 0.00942i)7-s + 0.333i·9-s − 0.561i·11-s + (−0.0525 + 0.0525i)13-s + (1.00 + 1.00i)17-s − 0.947·19-s − 0.00769·21-s + (0.663 + 0.663i)23-s + (0.136 − 0.136i)27-s − 0.451i·29-s − 0.633i·31-s + (−0.229 + 0.229i)33-s + (0.433 + 0.433i)37-s + 0.0429·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.484298229\)
\(L(\frac12)\) \(\approx\) \(1.484298229\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-0.0249 + 0.0249i)T - 7iT^{2} \)
11 \( 1 + 1.86iT - 11T^{2} \)
13 \( 1 + (0.189 - 0.189i)T - 13iT^{2} \)
17 \( 1 + (-4.14 - 4.14i)T + 17iT^{2} \)
19 \( 1 + 4.13T + 19T^{2} \)
23 \( 1 + (-3.18 - 3.18i)T + 23iT^{2} \)
29 \( 1 + 2.42iT - 29T^{2} \)
31 \( 1 + 3.52iT - 31T^{2} \)
37 \( 1 + (-2.63 - 2.63i)T + 37iT^{2} \)
41 \( 1 + 0.842T + 41T^{2} \)
43 \( 1 + (-4.40 - 4.40i)T + 43iT^{2} \)
47 \( 1 + (-2.87 + 2.87i)T - 47iT^{2} \)
53 \( 1 + (5.23 - 5.23i)T - 53iT^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + (-7.93 + 7.93i)T - 67iT^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 + (-2.53 + 2.53i)T - 73iT^{2} \)
79 \( 1 + 0.263T + 79T^{2} \)
83 \( 1 + (4.84 + 4.84i)T + 83iT^{2} \)
89 \( 1 - 4.85iT - 89T^{2} \)
97 \( 1 + (-12.2 - 12.2i)T + 97iT^{2} \)
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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

−8.892962326984633342190600220473, −8.008444668335127478830688565162, −7.56538763298692337604910815735, −6.40198064843006564101508679634, −6.01340363071943851362184591116, −5.10643272556729571163587715786, −4.11830390041821246116268190224, −3.17900413131033015185318169789, −2.00165385144991782911911560968, −0.858556758410327544473927499676, 0.75475172722960705110471459503, 2.22610130884078989779526642197, 3.27797056066519587871689020356, 4.26983299763179797485858318326, 5.04065257898307107729653057310, 5.70621331117393469218280553303, 6.77364190874391704319781545808, 7.25920928493887374279782073423, 8.328950202072422353641769827618, 9.003267215328959812835733052465

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