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Properties

Label 2-2e10-16.13-c1-0-7
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 + 2i)3-s + 5i·9-s + (−2 + 2i)11-s − 6·17-s + (6 + 6i)19-s + 5i·25-s + (−4 + 4i)27-s − 8·33-s + 6i·41-s + (6 − 6i)43-s + 7·49-s + (−12 − 12i)51-s + 24i·57-s + (10 − 10i)59-s + (−6 − 6i)67-s + ⋯
L(s)  = 1  + (1.15 + 1.15i)3-s + 1.66i·9-s + (−0.603 + 0.603i)11-s − 1.45·17-s + (1.37 + 1.37i)19-s + i·25-s + (−0.769 + 0.769i)27-s − 1.39·33-s + 0.937i·41-s + (0.914 − 0.914i)43-s + 49-s + (−1.68 − 1.68i)51-s + 3.17i·57-s + (1.30 − 1.30i)59-s + (−0.733 − 0.733i)67-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

\(L(1)\) \(\approx\) \(2.118156723\)
\(L(\frac12)\) \(\approx\) \(2.118156723\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 2i)T + 3iT^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (2 - 2i)T - 11iT^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + (-6 - 6i)T + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (-6 + 6i)T - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + (-10 + 10i)T - 59iT^{2} \)
61 \( 1 + 61iT^{2} \)
67 \( 1 + (6 + 6i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (2 + 2i)T + 83iT^{2} \)
89 \( 1 + 18iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.00167753879394501391472893078, −9.434506558798499245890033369067, −8.698799544335503279115229005473, −7.88182023823245553295176695194, −7.11827242659521147135499863243, −5.67865443344435722311545911789, −4.77111021245306703550164216763, −3.92442608162171711703254581100, −3.05371656727296707824483995781, −1.98773776291656032079881168238, 0.834582311387276827991493732713, 2.36445753953027619438674644962, 2.88459125440640351331937796670, 4.20231806608114379710959866625, 5.47715606407333095782087229301, 6.62717317503173937190731280993, 7.23827562527942798807709696167, 8.020284198885914417639769930380, 8.803849558941571265239170047156, 9.284251650252961596063019202131

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