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Properties

Label 2-2e10-16.13-c1-0-24
Degree $2$
Conductor $1024$
Sign $-0.923 - 0.382i$
Analytic cond. $8.17668$
Root an. cond. $2.85948$
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  + (−1.73 − 1.73i)3-s + (2.44 − 2.44i)5-s + 2.82i·7-s + 2.99i·9-s + (−1.73 + 1.73i)11-s + (−2.44 − 2.44i)13-s − 8.48·15-s − 4·17-s + (−1.73 − 1.73i)19-s + (4.89 − 4.89i)21-s − 2.82i·23-s − 6.99i·25-s + (−2.44 − 2.44i)29-s − 5.65·31-s + 5.99·33-s + ⋯
L(s)  = 1  + (−0.999 − 0.999i)3-s + (1.09 − 1.09i)5-s + 1.06i·7-s + 0.999i·9-s + (−0.522 + 0.522i)11-s + (−0.679 − 0.679i)13-s − 2.19·15-s − 0.970·17-s + (−0.397 − 0.397i)19-s + (1.06 − 1.06i)21-s − 0.589i·23-s − 1.39i·25-s + (−0.454 − 0.454i)29-s − 1.01·31-s + 1.04·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4475619670\)
\(L(\frac12)\) \(\approx\) \(0.4475619670\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (1.73 + 1.73i)T + 3iT^{2} \)
5 \( 1 + (-2.44 + 2.44i)T - 5iT^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 + (1.73 - 1.73i)T - 11iT^{2} \)
13 \( 1 + (2.44 + 2.44i)T + 13iT^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + (1.73 + 1.73i)T + 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (2.44 + 2.44i)T + 29iT^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (2.44 - 2.44i)T - 37iT^{2} \)
41 \( 1 + 2iT - 41T^{2} \)
43 \( 1 + (-8.66 + 8.66i)T - 43iT^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + (7.34 - 7.34i)T - 53iT^{2} \)
59 \( 1 + (1.73 - 1.73i)T - 59iT^{2} \)
61 \( 1 + (-2.44 - 2.44i)T + 61iT^{2} \)
67 \( 1 + (5.19 + 5.19i)T + 67iT^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 - 5.65T + 79T^{2} \)
83 \( 1 + (5.19 + 5.19i)T + 83iT^{2} \)
89 \( 1 - 8iT - 89T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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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

−9.335079598766173208496621632235, −8.786987532583072118518073708616, −7.72795504153010513122558325497, −6.75424093764081490177649401223, −5.88199787599454809401172232860, −5.38258551385324805840419757199, −4.66741728193073516666223126280, −2.45793613654299154185187427808, −1.76453183804338238368004738513, −0.21128616761450296225984751207, 1.99071906238451159329445946935, 3.35502516582054603110368915029, 4.37911896213165525816317887305, 5.26727619012495693439506293596, 6.14291453056439875809780022028, 6.79171291646951712325185029352, 7.69773353039140732519039558884, 9.216768388245465253283209046887, 9.833617681820679178779236349942, 10.48897365887088535039523633278

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