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Properties

Label 2-2e6-16.13-c23-0-1
Degree $2$
Conductor $64$
Sign $-0.535 + 0.844i$
Analytic cond. $214.530$
Root an. cond. $14.6468$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73e5 + 1.73e5i)3-s + (1.26e7 − 1.26e7i)5-s + 4.51e9i·7-s − 3.40e10i·9-s + (−9.96e11 + 9.96e11i)11-s + (−1.21e12 − 1.21e12i)13-s + 4.39e12·15-s + 1.64e14·17-s + (−4.04e14 − 4.04e14i)19-s + (−7.83e14 + 7.83e14i)21-s + 8.78e15i·23-s + 1.15e16i·25-s + (2.22e16 − 2.22e16i)27-s + (−5.23e15 − 5.23e15i)29-s − 6.76e16·31-s + ⋯
L(s)  = 1  + (0.564 + 0.564i)3-s + (0.116 − 0.116i)5-s + 0.863i·7-s − 0.361i·9-s + (−1.05 + 1.05i)11-s + (−0.187 − 0.187i)13-s + 0.131·15-s + 1.16·17-s + (−0.796 − 0.796i)19-s + (−0.488 + 0.488i)21-s + 1.92i·23-s + 0.973i·25-s + (0.769 − 0.769i)27-s + (−0.0796 − 0.0796i)29-s − 0.478·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.535 + 0.844i$
Analytic conductor: \(214.530\)
Root analytic conductor: \(14.6468\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :23/2),\ -0.535 + 0.844i)\)

Particular Values

\(L(12)\) \(\approx\) \(0.1548449774\)
\(L(\frac12)\) \(\approx\) \(0.1548449774\)
\(L(\frac{25}{2})\) 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 \( 1 + (-1.73e5 - 1.73e5i)T + 9.41e10iT^{2} \)
5 \( 1 + (-1.26e7 + 1.26e7i)T - 1.19e16iT^{2} \)
7 \( 1 - 4.51e9iT - 2.73e19T^{2} \)
11 \( 1 + (9.96e11 - 9.96e11i)T - 8.95e23iT^{2} \)
13 \( 1 + (1.21e12 + 1.21e12i)T + 4.17e25iT^{2} \)
17 \( 1 - 1.64e14T + 1.99e28T^{2} \)
19 \( 1 + (4.04e14 + 4.04e14i)T + 2.57e29iT^{2} \)
23 \( 1 - 8.78e15iT - 2.08e31T^{2} \)
29 \( 1 + (5.23e15 + 5.23e15i)T + 4.31e33iT^{2} \)
31 \( 1 + 6.76e16T + 2.00e34T^{2} \)
37 \( 1 + (-5.79e17 + 5.79e17i)T - 1.17e36iT^{2} \)
41 \( 1 - 1.77e18iT - 1.24e37T^{2} \)
43 \( 1 + (3.92e18 - 3.92e18i)T - 3.71e37iT^{2} \)
47 \( 1 + 2.61e19T + 2.87e38T^{2} \)
53 \( 1 + (-6.11e19 + 6.11e19i)T - 4.55e39iT^{2} \)
59 \( 1 + (-9.52e19 + 9.52e19i)T - 5.36e40iT^{2} \)
61 \( 1 + (1.94e20 + 1.94e20i)T + 1.15e41iT^{2} \)
67 \( 1 + (2.73e20 + 2.73e20i)T + 9.99e41iT^{2} \)
71 \( 1 - 3.21e21iT - 3.79e42T^{2} \)
73 \( 1 + 2.73e21iT - 7.18e42T^{2} \)
79 \( 1 + 2.70e21T + 4.42e43T^{2} \)
83 \( 1 + (8.93e21 + 8.93e21i)T + 1.37e44iT^{2} \)
89 \( 1 - 1.41e22iT - 6.85e44T^{2} \)
97 \( 1 + 1.02e23T + 4.96e45T^{2} \)
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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

−11.35231990585230249997822289314, −9.898299927363717005974151121454, −9.442673586646907404894567876274, −8.265492073568696130577915889411, −7.22266527929995380916293812339, −5.69213767108057555427719434737, −4.87701127863645826878396177782, −3.53503001138576474382606972967, −2.65142004103015024144906289718, −1.57046101309948508226425563248, 0.02512487394123841003548887410, 1.02786199784639796285524892826, 2.23891222236807042365127658341, 3.09865604014662874134869135102, 4.37185632112997466234675131909, 5.67147566147508336118656368940, 6.86701495398576919620823097935, 7.961171919884246236861132380841, 8.472075716441171161499812762073, 10.25365839775626743407106599390

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