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Properties

Label 2-69-69.68-c7-0-53
Degree $2$
Conductor $69$
Sign $0.992 + 0.124i$
Analytic cond. $21.5545$
Root an. cond. $4.64268$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 21.8i·2-s + (5.84 − 46.3i)3-s − 350.·4-s + (−1.01e3 − 127. i)6-s + 4.87e3i·8-s + (−2.11e3 − 542. i)9-s + (−2.05e3 + 1.62e4i)12-s + 451.·13-s + 6.18e4·16-s + (−1.18e4 + 4.63e4i)18-s + 5.83e4i·23-s + (2.26e5 + 2.85e4i)24-s − 7.81e4·25-s − 9.86e3i·26-s + (−3.75e4 + 9.51e4i)27-s + ⋯
L(s)  = 1  − 1.93i·2-s + (0.124 − 0.992i)3-s − 2.74·4-s + (−1.91 − 0.241i)6-s + 3.36i·8-s + (−0.968 − 0.247i)9-s + (−0.342 + 2.71i)12-s + 0.0569·13-s + 3.77·16-s + (−0.479 + 1.87i)18-s + 0.999i·23-s + (3.34 + 0.420i)24-s − 25-s − 0.110i·26-s + (−0.367 + 0.930i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.992 + 0.124i$
Analytic conductor: \(21.5545\)
Root analytic conductor: \(4.64268\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :7/2),\ 0.992 + 0.124i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0436587 - 0.00273876i\)
\(L(\frac12)\) \(\approx\) \(0.0436587 - 0.00273876i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.84 + 46.3i)T \)
23 \( 1 - 5.83e4iT \)
good2 \( 1 + 21.8iT - 128T^{2} \)
5 \( 1 + 7.81e4T^{2} \)
7 \( 1 - 8.23e5T^{2} \)
11 \( 1 + 1.94e7T^{2} \)
13 \( 1 - 451.T + 6.27e7T^{2} \)
17 \( 1 + 4.10e8T^{2} \)
19 \( 1 - 8.93e8T^{2} \)
29 \( 1 + 2.28e5iT - 1.72e10T^{2} \)
31 \( 1 + 2.18e5T + 2.75e10T^{2} \)
37 \( 1 - 9.49e10T^{2} \)
41 \( 1 - 8.76e5iT - 1.94e11T^{2} \)
43 \( 1 - 2.71e11T^{2} \)
47 \( 1 + 1.40e6iT - 5.06e11T^{2} \)
53 \( 1 + 1.17e12T^{2} \)
59 \( 1 - 3.15e6iT - 2.48e12T^{2} \)
61 \( 1 - 3.14e12T^{2} \)
67 \( 1 - 6.06e12T^{2} \)
71 \( 1 - 2.52e6iT - 9.09e12T^{2} \)
73 \( 1 + 5.43e6T + 1.10e13T^{2} \)
79 \( 1 - 1.92e13T^{2} \)
83 \( 1 + 2.71e13T^{2} \)
89 \( 1 + 4.42e13T^{2} \)
97 \( 1 - 8.07e13T^{2} \)
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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

−11.92546970746134920921370147912, −11.43394393610284210426831802424, −10.05274921379403225731466728986, −8.961107371313071058745695127194, −7.77045991892319958283206931884, −5.63486617284321988632240351081, −3.86839525236050465292411409734, −2.50750132032375092096440564448, −1.38183082205230075816305642799, −0.01615538672089723281827778351, 3.75387428883566289332722615820, 4.94850909938006679709741140218, 5.97102754789641152637948308714, 7.36124332668566680155562625534, 8.593426667082808884960590545801, 9.361204175216635651540966119500, 10.62346029894716538644570715708, 12.64509614667901675141919474997, 14.01812888748240365971053440307, 14.57904439108275310584307662981

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