Properties

Label 2-2e10-32.5-c1-0-3
Degree $2$
Conductor $1024$
Sign $-0.195 - 0.980i$
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  + (−0.942 − 2.27i)3-s + (2.49 + 1.03i)5-s + (−1.37 − 1.37i)7-s + (−2.17 + 2.17i)9-s + (−1.70 + 4.12i)11-s + (−4.86 + 2.01i)13-s − 6.66i·15-s + 6.31i·17-s + (−4.72 + 1.95i)19-s + (−1.82 + 4.41i)21-s + (0.288 − 0.288i)23-s + (1.63 + 1.63i)25-s + (0.159 + 0.0660i)27-s + (−0.428 − 1.03i)29-s − 3.69·31-s + ⋯
L(s)  = 1  + (−0.544 − 1.31i)3-s + (1.11 + 0.462i)5-s + (−0.518 − 0.518i)7-s + (−0.723 + 0.723i)9-s + (−0.515 + 1.24i)11-s + (−1.34 + 0.558i)13-s − 1.72i·15-s + 1.53i·17-s + (−1.08 + 0.449i)19-s + (−0.398 + 0.963i)21-s + (0.0602 − 0.0602i)23-s + (0.327 + 0.327i)25-s + (0.0306 + 0.0127i)27-s + (−0.0796 − 0.192i)29-s − 0.663·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.195 - 0.980i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4107574390\)
\(L(\frac12)\) \(\approx\) \(0.4107574390\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.942 + 2.27i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-2.49 - 1.03i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.37 + 1.37i)T + 7iT^{2} \)
11 \( 1 + (1.70 - 4.12i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (4.86 - 2.01i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 6.31iT - 17T^{2} \)
19 \( 1 + (4.72 - 1.95i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.288 + 0.288i)T - 23iT^{2} \)
29 \( 1 + (0.428 + 1.03i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 3.69T + 31T^{2} \)
37 \( 1 + (10.4 + 4.32i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.68 + 1.68i)T - 41iT^{2} \)
43 \( 1 + (1.67 - 4.03i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 4.83iT - 47T^{2} \)
53 \( 1 + (0.207 - 0.501i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-7.25 - 3.00i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.40 - 3.39i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-0.123 - 0.299i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-6.54 - 6.54i)T + 71iT^{2} \)
73 \( 1 + (-5.53 + 5.53i)T - 73iT^{2} \)
79 \( 1 + 0.877iT - 79T^{2} \)
83 \( 1 + (3.42 - 1.41i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-6.46 - 6.46i)T + 89iT^{2} \)
97 \( 1 + 3.58T + 97T^{2} \)
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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

−10.18441218468372614841581634990, −9.632009882136674959644593754224, −8.339011360507586326569430107098, −7.23796196256389435786670047830, −6.90039966180066826800857066102, −6.14577876641609311819066890247, −5.29096030762130034335470017337, −3.99692855400959001289683809147, −2.19864383347188602854870065886, −1.87727863060780599416240666501, 0.17791299672806548130507534457, 2.40336031278289267649206676461, 3.33851382786297530332748600636, 4.83697004945728386070075462351, 5.26631737185732204883598316898, 5.89576438974410221560097613773, 7.00319003833776903602079874588, 8.379050683809694495288281393266, 9.308634927313195414510142733440, 9.602444865566248008818352981827

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