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Properties

Label 2-2e10-16.13-c1-0-20
Degree $2$
Conductor $1024$
Sign $0.382 + 0.923i$
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  + (2.82 − 2.82i)5-s − 3i·9-s + (2.82 + 2.82i)13-s + 2·17-s − 11.0i·25-s + (−2.82 − 2.82i)29-s + (−8.48 + 8.48i)37-s − 10i·41-s + (−8.48 − 8.48i)45-s + 7·49-s + (−2.82 + 2.82i)53-s + (8.48 + 8.48i)61-s + 16.0·65-s − 6i·73-s − 9·81-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)5-s − i·9-s + (0.784 + 0.784i)13-s + 0.485·17-s − 2.20i·25-s + (−0.525 − 0.525i)29-s + (−1.39 + 1.39i)37-s − 1.56i·41-s + (−1.26 − 1.26i)45-s + 49-s + (−0.388 + 0.388i)53-s + (1.08 + 1.08i)61-s + 1.98·65-s − 0.702i·73-s − 81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.382 + 0.923i$
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: \((2,\ 1024,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.037042393\)
\(L(\frac12)\) \(\approx\) \(2.037042393\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT^{2} \)
5 \( 1 + (-2.82 + 2.82i)T - 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11iT^{2} \)
13 \( 1 + (-2.82 - 2.82i)T + 13iT^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (2.82 + 2.82i)T + 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (8.48 - 8.48i)T - 37iT^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (2.82 - 2.82i)T - 53iT^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (-8.48 - 8.48i)T + 61iT^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 10iT - 89T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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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

−9.703275999072454986569375231145, −8.872559472518868588079430028459, −8.602773449742420048004251352876, −7.14586328455134446352755337421, −6.16412383270172508996917318555, −5.61539393165454219855814184631, −4.59354653671398767727413174612, −3.56948708546434868197597579258, −1.99892333399940755888540440435, −1.00574272082875459696307284961, 1.69087459122260463322952877064, 2.68017699394703907603747882189, 3.60979273820762860950997702900, 5.22980056795375191386541686254, 5.77145181090535733330274541279, 6.69639240747841423715517868433, 7.49411578571984833129837680311, 8.412932550411099968127786977725, 9.483706590324607110963825750688, 10.23569053622082086164273179807

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