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Properties

Label 2-2e10-16.5-c1-0-1
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  + (−1.41 + 1.41i)3-s + (−1.41 − 1.41i)5-s + 4i·7-s − 1.00i·9-s + (1.41 + 1.41i)11-s + (−1.41 + 1.41i)13-s + 4.00·15-s + 2·17-s + (1.41 − 1.41i)19-s + (−5.65 − 5.65i)21-s + 4i·23-s − 0.999i·25-s + (−2.82 − 2.82i)27-s + (−4.24 + 4.24i)29-s − 4.00·33-s + ⋯
L(s)  = 1  + (−0.816 + 0.816i)3-s + (−0.632 − 0.632i)5-s + 1.51i·7-s − 0.333i·9-s + (0.426 + 0.426i)11-s + (−0.392 + 0.392i)13-s + 1.03·15-s + 0.485·17-s + (0.324 − 0.324i)19-s + (−1.23 − 1.23i)21-s + 0.834i·23-s − 0.199i·25-s + (−0.544 − 0.544i)27-s + (−0.787 + 0.787i)29-s − 0.696·33-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: $\chi_{1024} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4023406400\)
\(L(\frac12)\) \(\approx\) \(0.4023406400\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 1.41i)T - 3iT^{2} \)
5 \( 1 + (1.41 + 1.41i)T + 5iT^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + (-1.41 - 1.41i)T + 11iT^{2} \)
13 \( 1 + (1.41 - 1.41i)T - 13iT^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + (-1.41 + 1.41i)T - 19iT^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + (4.24 - 4.24i)T - 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (7.07 + 7.07i)T + 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (4.24 + 4.24i)T + 43iT^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (4.24 + 4.24i)T + 53iT^{2} \)
59 \( 1 + (-9.89 - 9.89i)T + 59iT^{2} \)
61 \( 1 + (-1.41 + 1.41i)T - 61iT^{2} \)
67 \( 1 + (7.07 - 7.07i)T - 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (4.24 - 4.24i)T - 83iT^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.37766481314547708102252187507, −9.486702918089384720247225049366, −8.978566805269940543189893531142, −8.054056165131917240853769935424, −7.00880621959966076447059244682, −5.79554414405538721509947445420, −5.21100749556451135527477347663, −4.52070265912957728645985008856, −3.40427206199065111265939285842, −1.89091151206295870206328562803, 0.21513376187048329687215324420, 1.37037615342815881249183204910, 3.24030806216906782130917264120, 3.99698149468041112019614257119, 5.24456934365175119541046183114, 6.33061058516496070954588142833, 6.99761075894000849361771495442, 7.51332861988763334716537874738, 8.331534416965712938364822837319, 9.779011045846737118245375800796

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