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Properties

Label 2-2e6-16.13-c21-0-11
Degree $2$
Conductor $64$
Sign $0.788 + 0.615i$
Analytic cond. $178.865$
Root an. cond. $13.3740$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.09e5 − 1.09e5i)3-s + (−1.97e7 + 1.97e7i)5-s − 1.02e9i·7-s + 1.36e10i·9-s + (7.32e10 − 7.32e10i)11-s + (−4.28e11 − 4.28e11i)13-s + 4.32e12·15-s + 6.62e12·17-s + (1.48e13 + 1.48e13i)19-s + (−1.12e14 + 1.12e14i)21-s − 2.73e14i·23-s − 3.01e14i·25-s + (3.46e14 − 3.46e14i)27-s + (1.70e15 + 1.70e15i)29-s − 1.42e15·31-s + ⋯
L(s)  = 1  + (−1.07 − 1.07i)3-s + (−0.903 + 0.903i)5-s − 1.36i·7-s + 1.30i·9-s + (0.851 − 0.851i)11-s + (−0.861 − 0.861i)13-s + 1.93·15-s + 0.796·17-s + (0.557 + 0.557i)19-s + (−1.46 + 1.46i)21-s − 1.37i·23-s − 0.632i·25-s + (0.323 − 0.323i)27-s + (0.754 + 0.754i)29-s − 0.311·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(178.865\)
Root analytic conductor: \(13.3740\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :21/2),\ 0.788 + 0.615i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.9049095791\)
\(L(\frac12)\) \(\approx\) \(0.9049095791\)
\(L(\frac{23}{2})\) 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 \( 1 + (1.09e5 + 1.09e5i)T + 1.04e10iT^{2} \)
5 \( 1 + (1.97e7 - 1.97e7i)T - 4.76e14iT^{2} \)
7 \( 1 + 1.02e9iT - 5.58e17T^{2} \)
11 \( 1 + (-7.32e10 + 7.32e10i)T - 7.40e21iT^{2} \)
13 \( 1 + (4.28e11 + 4.28e11i)T + 2.47e23iT^{2} \)
17 \( 1 - 6.62e12T + 6.90e25T^{2} \)
19 \( 1 + (-1.48e13 - 1.48e13i)T + 7.14e26iT^{2} \)
23 \( 1 + 2.73e14iT - 3.94e28T^{2} \)
29 \( 1 + (-1.70e15 - 1.70e15i)T + 5.13e30iT^{2} \)
31 \( 1 + 1.42e15T + 2.08e31T^{2} \)
37 \( 1 + (9.05e15 - 9.05e15i)T - 8.55e32iT^{2} \)
41 \( 1 + 5.53e16iT - 7.38e33T^{2} \)
43 \( 1 + (-8.48e16 + 8.48e16i)T - 2.00e34iT^{2} \)
47 \( 1 + 5.38e17T + 1.30e35T^{2} \)
53 \( 1 + (6.11e17 - 6.11e17i)T - 1.62e36iT^{2} \)
59 \( 1 + (5.01e18 - 5.01e18i)T - 1.54e37iT^{2} \)
61 \( 1 + (-7.57e18 - 7.57e18i)T + 3.10e37iT^{2} \)
67 \( 1 + (-1.45e19 - 1.45e19i)T + 2.22e38iT^{2} \)
71 \( 1 - 3.61e19iT - 7.52e38T^{2} \)
73 \( 1 - 1.74e19iT - 1.34e39T^{2} \)
79 \( 1 + 9.41e18T + 7.08e39T^{2} \)
83 \( 1 + (-1.22e20 - 1.22e20i)T + 1.99e40iT^{2} \)
89 \( 1 - 4.34e20iT - 8.65e40T^{2} \)
97 \( 1 + 4.29e20T + 5.27e41T^{2} \)
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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

−10.93741647095069390171091056403, −10.24335909052124668390310372775, −8.107695608409419069135268252948, −7.22097354831048361321496083800, −6.72298284754645623078445112752, −5.49094276567862070308124626652, −3.99294122882276598297267398472, −2.98558089424756271550095832779, −1.15321587223904079739455286147, −0.61631046027350591499038414702, 0.37044949649305891067220546592, 1.79001174273617372461025966119, 3.51825285881858951722084306098, 4.71590691210196778274339048403, 5.06257951903042906870635592083, 6.34083955245928145477076464067, 7.84681790035594903183819046536, 9.327907747401184060349901941577, 9.628233188066047873244409303867, 11.47111457847857672069342072657

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