Properties

Label 2-1280-40.19-c2-0-21
Degree $2$
Conductor $1280$
Sign $-0.707 - 0.707i$
Analytic cond. $34.8774$
Root an. cond. $5.90571$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·3-s − 5i·5-s − 4·7-s − 7·9-s + 20·15-s − 16i·21-s + 44·23-s − 25·25-s + 8i·27-s + 22i·29-s + 20i·35-s − 62·41-s + 76i·43-s + 35i·45-s + 4·47-s + ⋯
L(s)  = 1  + 1.33i·3-s i·5-s − 0.571·7-s − 0.777·9-s + 1.33·15-s − 0.761i·21-s + 1.91·23-s − 25-s + 0.296i·27-s + 0.758i·29-s + 0.571i·35-s − 1.51·41-s + 1.76i·43-s + 0.777i·45-s + 0.0851·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(34.8774\)
Root analytic conductor: \(5.90571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.218667633\)
\(L(\frac12)\) \(\approx\) \(1.218667633\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 5iT \)
good3 \( 1 - 4iT - 9T^{2} \)
7 \( 1 + 4T + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 + 361T^{2} \)
23 \( 1 - 44T + 529T^{2} \)
29 \( 1 - 22iT - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + 1.36e3T^{2} \)
41 \( 1 + 62T + 1.68e3T^{2} \)
43 \( 1 - 76iT - 1.84e3T^{2} \)
47 \( 1 - 4T + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 - 58iT - 3.72e3T^{2} \)
67 \( 1 - 116iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 76iT - 6.88e3T^{2} \)
89 \( 1 - 142T + 7.92e3T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−9.709522022090276329098563625636, −9.068973803963312490243754942738, −8.564704776049152732813003457580, −7.38110465145316162593747304437, −6.33636104148734428505017865299, −5.18702157200992367309547424656, −4.80844599379100670912543286831, −3.80199478947581832955197409032, −2.96178156975455524157690426332, −1.21527649783633792059533219864, 0.37852056963888892286509842834, 1.77674798588697002448546208097, 2.76581517558307102902251633139, 3.61863463185476116642467749967, 5.11922337529695829585761535868, 6.24863816301040650669559869645, 6.75858471785717614873811513257, 7.32895438964099767780841807848, 8.129430215047166191015492150163, 9.123698712108264881233629007433

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