Properties

Label 2-1280-20.3-c1-0-29
Degree $2$
Conductor $1280$
Sign $0.725 + 0.688i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
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  + (0.456 − 0.456i)3-s + (0.456 − 2.18i)5-s + (2.79 + 2.79i)7-s + 2.58i·9-s − 4.37i·11-s + (−1.73 − 1.73i)13-s + (−0.791 − 1.20i)15-s + (3 − 3i)17-s + 3.46·19-s + 2.55·21-s + (0.791 − 0.791i)23-s + (−4.58 − 1.99i)25-s + (2.55 + 2.55i)27-s + 5.29i·29-s − 1.58i·31-s + ⋯
L(s)  = 1  + (0.263 − 0.263i)3-s + (0.204 − 0.978i)5-s + (1.05 + 1.05i)7-s + 0.860i·9-s − 1.31i·11-s + (−0.480 − 0.480i)13-s + (−0.204 − 0.312i)15-s + (0.727 − 0.727i)17-s + 0.794·19-s + 0.556·21-s + (0.164 − 0.164i)23-s + (−0.916 − 0.399i)25-s + (0.490 + 0.490i)27-s + 0.982i·29-s − 0.284i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.725 + 0.688i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.725 + 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.148671310\)
\(L(\frac12)\) \(\approx\) \(2.148671310\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.456 + 2.18i)T \)
good3 \( 1 + (-0.456 + 0.456i)T - 3iT^{2} \)
7 \( 1 + (-2.79 - 2.79i)T + 7iT^{2} \)
11 \( 1 + 4.37iT - 11T^{2} \)
13 \( 1 + (1.73 + 1.73i)T + 13iT^{2} \)
17 \( 1 + (-3 + 3i)T - 17iT^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + (-0.791 + 0.791i)T - 23iT^{2} \)
29 \( 1 - 5.29iT - 29T^{2} \)
31 \( 1 + 1.58iT - 31T^{2} \)
37 \( 1 + (5.19 - 5.19i)T - 37iT^{2} \)
41 \( 1 - 7.58T + 41T^{2} \)
43 \( 1 + (-8.29 + 8.29i)T - 43iT^{2} \)
47 \( 1 + (-0.791 - 0.791i)T + 47iT^{2} \)
53 \( 1 + (-2.64 - 2.64i)T + 53iT^{2} \)
59 \( 1 - 5.29T + 59T^{2} \)
61 \( 1 + 9.66T + 61T^{2} \)
67 \( 1 + (8.29 + 8.29i)T + 67iT^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + (0.582 + 0.582i)T + 73iT^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + (-4.83 + 4.83i)T - 83iT^{2} \)
89 \( 1 + 3.16iT - 89T^{2} \)
97 \( 1 + (-0.582 + 0.582i)T - 97iT^{2} \)
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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

−9.191793173131798170026398689841, −8.803891205178774141331437885052, −7.963255473107796436289823434514, −7.52568625546845865005178075160, −5.89933965757896409529912349584, −5.29109859685167791488360434252, −4.80991315529047471813027697613, −3.19322040151563065841524532809, −2.19673399087993089553681857409, −1.03134706760424732809863548509, 1.36404994139170662485891913191, 2.56866258472619128966459326508, 3.83898479143446412870558235589, 4.36759794133479329447175130670, 5.57153937644792407486502800265, 6.67543628423932612868921559487, 7.40695722356328192417620332022, 7.83858552861095712432919055794, 9.194358681474842354433933408693, 9.858837318533893795554051827755

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