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Properties

Label 2-1440-160.59-c0-0-0
Degree $2$
Conductor $1440$
Sign $0.555 - 0.831i$
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  + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (−0.382 + 0.923i)5-s + (0.923 + 0.382i)8-s + 10-s − i·16-s − 1.84·17-s + (0.707 + 1.70i)19-s + (−0.382 − 0.923i)20-s + (−0.541 + 0.541i)23-s + (−0.707 − 0.707i)25-s + 1.41i·31-s + (−0.923 + 0.382i)32-s + (0.707 + 1.70i)34-s + (1.30 − 1.30i)38-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (−0.382 + 0.923i)5-s + (0.923 + 0.382i)8-s + 10-s − i·16-s − 1.84·17-s + (0.707 + 1.70i)19-s + (−0.382 − 0.923i)20-s + (−0.541 + 0.541i)23-s + (−0.707 − 0.707i)25-s + 1.41i·31-s + (−0.923 + 0.382i)32-s + (0.707 + 1.70i)34-s + (1.30 − 1.30i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :0),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5821474122\)
\(L(\frac12)\) \(\approx\) \(0.5821474122\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 0.923i)T \)
3 \( 1 \)
5 \( 1 + (0.382 - 0.923i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + 1.84T + T^{2} \)
19 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 - 1.84iT - T^{2} \)
53 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−10.09187776607255962188728958921, −9.146281814243592600700810587604, −8.349983800570463803223003904208, −7.57945159767021257871677235108, −6.80734525506907593971899255670, −5.71363721891925924303653795177, −4.43294292158832820538316341277, −3.68305557050048280982572915621, −2.77684876074524710384355243123, −1.69835353939869759421523536021, 0.54445633806786032097776000841, 2.21252218583247207251733591237, 4.02122600347840540955450244112, 4.67775648496282168759961560636, 5.44307911576791055408127667600, 6.49512367198417681529190376163, 7.18698299468649149958441687349, 8.052026828292500374908676284591, 8.849581699687844513189507126549, 9.192149280138214790639383854449

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