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Properties

Label 2-2e6-16.13-c23-0-10
Degree $2$
Conductor $64$
Sign $0.688 - 0.724i$
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  + (−6.84e4 − 6.84e4i)3-s + (−2.70e6 + 2.70e6i)5-s + 2.28e9i·7-s − 8.47e10i·9-s + (4.33e11 − 4.33e11i)11-s + (−3.60e12 − 3.60e12i)13-s + 3.70e11·15-s − 2.43e14·17-s + (9.90e12 + 9.90e12i)19-s + (1.56e14 − 1.56e14i)21-s + 3.84e15i·23-s + 1.19e16i·25-s + (−1.22e16 + 1.22e16i)27-s + (−4.39e16 − 4.39e16i)29-s + 1.06e17·31-s + ⋯
L(s)  = 1  + (−0.223 − 0.223i)3-s + (−0.0247 + 0.0247i)5-s + 0.436i·7-s − 0.900i·9-s + (0.457 − 0.457i)11-s + (−0.557 − 0.557i)13-s + 0.0110·15-s − 1.72·17-s + (0.0195 + 0.0195i)19-s + (0.0972 − 0.0972i)21-s + 0.840i·23-s + 0.998i·25-s + (−0.423 + 0.423i)27-s + (−0.668 − 0.668i)29-s + 0.755·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.724i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (0.688 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.688 - 0.724i$
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: \((2,\ 64,\ (\ :23/2),\ 0.688 - 0.724i)\)

Particular Values

\(L(12)\) \(\approx\) \(1.013002537\)
\(L(\frac12)\) \(\approx\) \(1.013002537\)
\(L(\frac{25}{2})\) 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 \( 1 + (6.84e4 + 6.84e4i)T + 9.41e10iT^{2} \)
5 \( 1 + (2.70e6 - 2.70e6i)T - 1.19e16iT^{2} \)
7 \( 1 - 2.28e9iT - 2.73e19T^{2} \)
11 \( 1 + (-4.33e11 + 4.33e11i)T - 8.95e23iT^{2} \)
13 \( 1 + (3.60e12 + 3.60e12i)T + 4.17e25iT^{2} \)
17 \( 1 + 2.43e14T + 1.99e28T^{2} \)
19 \( 1 + (-9.90e12 - 9.90e12i)T + 2.57e29iT^{2} \)
23 \( 1 - 3.84e15iT - 2.08e31T^{2} \)
29 \( 1 + (4.39e16 + 4.39e16i)T + 4.31e33iT^{2} \)
31 \( 1 - 1.06e17T + 2.00e34T^{2} \)
37 \( 1 + (-1.78e17 + 1.78e17i)T - 1.17e36iT^{2} \)
41 \( 1 + 1.48e18iT - 1.24e37T^{2} \)
43 \( 1 + (-4.74e18 + 4.74e18i)T - 3.71e37iT^{2} \)
47 \( 1 + 1.55e19T + 2.87e38T^{2} \)
53 \( 1 + (2.36e19 - 2.36e19i)T - 4.55e39iT^{2} \)
59 \( 1 + (1.81e20 - 1.81e20i)T - 5.36e40iT^{2} \)
61 \( 1 + (4.63e20 + 4.63e20i)T + 1.15e41iT^{2} \)
67 \( 1 + (-9.87e20 - 9.87e20i)T + 9.99e41iT^{2} \)
71 \( 1 - 3.79e20iT - 3.79e42T^{2} \)
73 \( 1 + 2.17e21iT - 7.18e42T^{2} \)
79 \( 1 + 4.95e21T + 4.42e43T^{2} \)
83 \( 1 + (6.24e21 + 6.24e21i)T + 1.37e44iT^{2} \)
89 \( 1 - 3.61e22iT - 6.85e44T^{2} \)
97 \( 1 - 1.92e21T + 4.96e45T^{2} \)
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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

−11.00718820811482419562505409877, −9.539130162365164521603973022380, −8.820685478274983891925107498303, −7.45382488044647542273328167816, −6.41324985987941413720193835587, −5.51995494447843763254499165861, −4.17134236272855731013827074518, −3.06187456076921755554757856618, −1.86648432080994102308505601573, −0.65165585868574931699479556371, 0.26256582730915568792503589807, 1.70817214363753110485060862221, 2.61627756587535910032669354962, 4.32598723572376184622289162259, 4.70140557652301584058134131175, 6.29154915296582584928405733747, 7.19432370062688086778554132183, 8.397669953384230470257940821841, 9.539536774927303171948918943518, 10.59488211113998257686290055770

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