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Properties

Label 2-69-3.2-c6-0-19
Degree $2$
Conductor $69$
Sign $-0.359 - 0.933i$
Analytic cond. $15.8737$
Root an. cond. $3.98418$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.4i·2-s + (25.1 − 9.70i)3-s − 143.·4-s − 103. i·5-s + (139. + 362. i)6-s + 470.·7-s − 1.14e3i·8-s + (540. − 488. i)9-s + 1.49e3·10-s + 2.59e3i·11-s + (−3.61e3 + 1.39e3i)12-s − 326.·13-s + 6.78e3i·14-s + (−1.00e3 − 2.61e3i)15-s + 7.29e3·16-s + 1.60e3i·17-s + ⋯
L(s)  = 1  + 1.80i·2-s + (0.933 − 0.359i)3-s − 2.24·4-s − 0.831i·5-s + (0.647 + 1.68i)6-s + 1.37·7-s − 2.23i·8-s + (0.741 − 0.670i)9-s + 1.49·10-s + 1.94i·11-s + (−2.09 + 0.805i)12-s − 0.148·13-s + 2.47i·14-s + (−0.298 − 0.775i)15-s + 1.78·16-s + 0.326i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 - 0.933i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.359 - 0.933i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.359 - 0.933i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.47280 + 2.14547i\)
\(L(\frac12)\) \(\approx\) \(1.47280 + 2.14547i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-25.1 + 9.70i)T \)
23 \( 1 - 2.53e3iT \)
good2 \( 1 - 14.4iT - 64T^{2} \)
5 \( 1 + 103. iT - 1.56e4T^{2} \)
7 \( 1 - 470.T + 1.17e5T^{2} \)
11 \( 1 - 2.59e3iT - 1.77e6T^{2} \)
13 \( 1 + 326.T + 4.82e6T^{2} \)
17 \( 1 - 1.60e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.31e4T + 4.70e7T^{2} \)
29 \( 1 - 9.23e3iT - 5.94e8T^{2} \)
31 \( 1 + 3.17e4T + 8.87e8T^{2} \)
37 \( 1 - 5.89e4T + 2.56e9T^{2} \)
41 \( 1 + 6.69e4iT - 4.75e9T^{2} \)
43 \( 1 + 8.27e4T + 6.32e9T^{2} \)
47 \( 1 + 2.51e4iT - 1.07e10T^{2} \)
53 \( 1 - 2.14e5iT - 2.21e10T^{2} \)
59 \( 1 + 4.90e4iT - 4.21e10T^{2} \)
61 \( 1 - 9.00e4T + 5.15e10T^{2} \)
67 \( 1 + 3.15e5T + 9.04e10T^{2} \)
71 \( 1 + 2.60e4iT - 1.28e11T^{2} \)
73 \( 1 - 2.35e4T + 1.51e11T^{2} \)
79 \( 1 + 3.87e5T + 2.43e11T^{2} \)
83 \( 1 + 5.48e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.70e5iT - 4.96e11T^{2} \)
97 \( 1 + 6.08e5T + 8.32e11T^{2} \)
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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

−14.30993499524803830320620319494, −13.20433676717160040961530838741, −12.16828512114811498911194215101, −9.677838092145930537187676968453, −8.812094268064714964582067935775, −7.70351359093910606349121235125, −7.23058544352816275829715438496, −5.24425409538510783391128227117, −4.39393582486599059636196173439, −1.47455780281066284152540484274, 1.18012819418328858720799529899, 2.70777483168125693656454227671, 3.54715929405032254059944644234, 5.07699345663222054258878228598, 7.84649186435112174741391043296, 8.860428283715320054619322419101, 10.01504388928993171491648074855, 11.16153656574013875674555299139, 11.47968461162900045173386686652, 13.30138183214661275077358510644

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