Properties

Label 2-1440-40.37-c0-0-0
Degree $2$
Conductor $1440$
Sign $0.850 - 0.525i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
Motivic weight $0$
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)5-s + (1 + i)7-s + 1.41i·11-s + 1.00i·25-s + 1.41·29-s − 1.41i·35-s + i·49-s + (1.00 − 1.00i)55-s + 1.41·59-s + (1 − i)73-s + (−1.41 + 1.41i)77-s + (−1.41 − 1.41i)83-s + (−1 − i)97-s + 1.41i·101-s + (−1 + i)103-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s + (1 + i)7-s + 1.41i·11-s + 1.00i·25-s + 1.41·29-s − 1.41i·35-s + i·49-s + (1.00 − 1.00i)55-s + 1.41·59-s + (1 − i)73-s + (−1.41 + 1.41i)77-s + (−1.41 − 1.41i)83-s + (−1 − i)97-s + 1.41i·101-s + (−1 + i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066008454\)
\(L(\frac12)\) \(\approx\) \(1.066008454\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-1 - i)T + iT^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−9.658632169596807141687616371558, −8.870154338453486703574451614496, −8.230681187904923458885849734479, −7.57354930504834081322335695759, −6.62240641491018134764554524249, −5.36110192252952478220764486783, −4.82763904591644992128280019524, −4.05074312259481213836843855944, −2.57850643967214927653842104658, −1.51646690258066704947517234067, 1.01192412048972559087686914864, 2.69607859618904268413154831238, 3.69127556985221512978170843970, 4.42053086846019867613949769744, 5.49478325187509058579704908580, 6.57920763497318230169131748881, 7.23877932339360326864512391072, 8.234703536003106443540570371145, 8.406482359254169469636465008741, 9.835613993261264500516595646176

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