Properties

Label 2-2e10-16.13-c1-0-17
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: $\chi_{1024} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.345013099\)
\(L(\frac12)\) \(\approx\) \(2.345013099\)
\(L(\frac{3}{2})\) 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

−9.936203449858407066101766210083, −9.105151362839531963921654237869, −8.474509505206986133625904796930, −7.45948193348215694965451488623, −6.62333895088324518299342854291, −5.35633980321687366441404325791, −4.37316585694665343858710972664, −3.83788875370598494527992325940, −2.56983300073975949157135885651, −1.06757726028612270039533786595, 1.63751757047662800139132892455, 2.65285222605664471768876986082, 3.11869155992713791672063183052, 4.94674293624403602174610833580, 6.04929189231026863666023906150, 6.37714583895726910445856578997, 7.919510699779543160589495179527, 8.056407383876123896830699747140, 9.098806193804102531797053295252, 9.831823326973790446938575509174

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