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Properties

Label 2-2e10-32.13-c1-0-22
Degree $2$
Conductor $1024$
Sign $0.980 + 0.195i$
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  + (−0.401 + 0.969i)3-s + (3.49 − 1.44i)5-s + (3.21 − 3.21i)7-s + (1.34 + 1.34i)9-s + (0.363 + 0.878i)11-s + (2.37 + 0.985i)13-s + 3.97i·15-s − 1.34i·17-s + (−3.10 − 1.28i)19-s + (1.82 + 4.41i)21-s + (−4.30 − 4.30i)23-s + (6.60 − 6.60i)25-s + (−4.74 + 1.96i)27-s + (−0.600 + 1.44i)29-s − 3.69·31-s + ⋯
L(s)  = 1  + (−0.231 + 0.559i)3-s + (1.56 − 0.648i)5-s + (1.21 − 1.21i)7-s + (0.447 + 0.447i)9-s + (0.109 + 0.264i)11-s + (0.659 + 0.273i)13-s + 1.02i·15-s − 0.326i·17-s + (−0.712 − 0.295i)19-s + (0.398 + 0.963i)21-s + (−0.896 − 0.896i)23-s + (1.32 − 1.32i)25-s + (−0.914 + 0.378i)27-s + (−0.111 + 0.269i)29-s − 0.663·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.319421740\)
\(L(\frac12)\) \(\approx\) \(2.319421740\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.401 - 0.969i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-3.49 + 1.44i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-3.21 + 3.21i)T - 7iT^{2} \)
11 \( 1 + (-0.363 - 0.878i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-2.37 - 0.985i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 1.34iT - 17T^{2} \)
19 \( 1 + (3.10 + 1.28i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.30 + 4.30i)T + 23iT^{2} \)
29 \( 1 + (0.600 - 1.44i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 3.69T + 31T^{2} \)
37 \( 1 + (4.03 - 1.67i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (5.34 + 5.34i)T + 41iT^{2} \)
43 \( 1 + (-3.01 - 7.27i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 1.02iT - 47T^{2} \)
53 \( 1 + (-2.69 - 6.49i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (5.90 - 2.44i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.83 - 4.42i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (2.58 - 6.23i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-5.75 + 5.75i)T - 71iT^{2} \)
73 \( 1 + (2.94 + 2.94i)T + 73iT^{2} \)
79 \( 1 - 15.4iT - 79T^{2} \)
83 \( 1 + (-11.2 - 4.66i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (9.04 - 9.04i)T - 89iT^{2} \)
97 \( 1 - 8.41T + 97T^{2} \)
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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

−10.09896795751356460840143645336, −9.191550497168641035772220016504, −8.411077523077637426744787989490, −7.40436250460151639591487856890, −6.43157960539404566166363843259, −5.40615509640993197841184971493, −4.67597689222913875247277759421, −4.08094889157500311175804581319, −2.09354339487738476623999640649, −1.31105966343929282618981484521, 1.68980808901866865006814343539, 2.04409572115960981646456515340, 3.54943280032473707821867624182, 5.09370261886388058772378731519, 5.97442785212152204007521506353, 6.20881147214360494003406739484, 7.39563653071674031059872025476, 8.400370422478400718201628402097, 9.123460785008127450180597036278, 9.987263487677573306061128371504

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