Properties

Label 2-69-23.11-c6-0-22
Degree $2$
Conductor $69$
Sign $-0.746 + 0.665i$
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  + (11.8 − 7.59i)2-s + (2.21 − 15.4i)3-s + (55.4 − 121. i)4-s + (−0.763 − 2.60i)5-s + (−91.0 − 199. i)6-s + (353. − 306. i)7-s + (−139. − 967. i)8-s + (−233. − 68.4i)9-s + (−28.7 − 24.9i)10-s + (−1.16e3 + 1.82e3i)11-s + (−1.75e3 − 1.12e3i)12-s + (1.99e3 − 2.29e3i)13-s + (1.85e3 − 6.31e3i)14-s + (−41.8 + 6.01i)15-s + (−3.40e3 − 3.92e3i)16-s + (−6.42e3 + 2.93e3i)17-s + ⋯
L(s)  = 1  + (1.47 − 0.949i)2-s + (0.0821 − 0.571i)3-s + (0.866 − 1.89i)4-s + (−0.00611 − 0.0208i)5-s + (−0.421 − 0.922i)6-s + (1.03 − 0.893i)7-s + (−0.271 − 1.89i)8-s + (−0.319 − 0.0939i)9-s + (−0.0287 − 0.0249i)10-s + (−0.878 + 1.36i)11-s + (−1.01 − 0.651i)12-s + (0.906 − 1.04i)13-s + (0.675 − 2.30i)14-s + (−0.0123 + 0.00178i)15-s + (−0.830 − 0.958i)16-s + (−1.30 + 0.597i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.746 + 0.665i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.58146 - 4.14898i\)
\(L(\frac12)\) \(\approx\) \(1.58146 - 4.14898i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.21 + 15.4i)T \)
23 \( 1 + (6.04e3 + 1.05e4i)T \)
good2 \( 1 + (-11.8 + 7.59i)T + (26.5 - 58.2i)T^{2} \)
5 \( 1 + (0.763 + 2.60i)T + (-1.31e4 + 8.44e3i)T^{2} \)
7 \( 1 + (-353. + 306. i)T + (1.67e4 - 1.16e5i)T^{2} \)
11 \( 1 + (1.16e3 - 1.82e3i)T + (-7.35e5 - 1.61e6i)T^{2} \)
13 \( 1 + (-1.99e3 + 2.29e3i)T + (-6.86e5 - 4.77e6i)T^{2} \)
17 \( 1 + (6.42e3 - 2.93e3i)T + (1.58e7 - 1.82e7i)T^{2} \)
19 \( 1 + (-2.15e3 - 984. i)T + (3.08e7 + 3.55e7i)T^{2} \)
29 \( 1 + (-1.98e4 - 4.34e4i)T + (-3.89e8 + 4.49e8i)T^{2} \)
31 \( 1 + (-1.40e3 - 9.78e3i)T + (-8.51e8 + 2.50e8i)T^{2} \)
37 \( 1 + (-8.35e3 + 2.84e4i)T + (-2.15e9 - 1.38e9i)T^{2} \)
41 \( 1 + (-5.20e4 + 1.52e4i)T + (3.99e9 - 2.56e9i)T^{2} \)
43 \( 1 + (1.25e4 + 1.81e3i)T + (6.06e9 + 1.78e9i)T^{2} \)
47 \( 1 - 8.39e4T + 1.07e10T^{2} \)
53 \( 1 + (3.18e4 - 2.75e4i)T + (3.15e9 - 2.19e10i)T^{2} \)
59 \( 1 + (-1.61e5 + 1.86e5i)T + (-6.00e9 - 4.17e10i)T^{2} \)
61 \( 1 + (1.18e5 - 1.70e4i)T + (4.94e10 - 1.45e10i)T^{2} \)
67 \( 1 + (-5.12e4 - 7.97e4i)T + (-3.75e10 + 8.22e10i)T^{2} \)
71 \( 1 + (-2.22e4 + 1.42e4i)T + (5.32e10 - 1.16e11i)T^{2} \)
73 \( 1 + (1.12e5 - 2.45e5i)T + (-9.91e10 - 1.14e11i)T^{2} \)
79 \( 1 + (-6.14e5 - 5.32e5i)T + (3.45e10 + 2.40e11i)T^{2} \)
83 \( 1 + (9.80e4 - 3.33e5i)T + (-2.75e11 - 1.76e11i)T^{2} \)
89 \( 1 + (9.57e5 + 1.37e5i)T + (4.76e11 + 1.40e11i)T^{2} \)
97 \( 1 + (-1.62e5 - 5.54e5i)T + (-7.00e11 + 4.50e11i)T^{2} \)
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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

−12.91373379035812018055786589981, −12.41934743312730880744152276010, −10.84574400970815050293473449878, −10.55898408937949518159644985576, −8.217504091886182123575679378563, −6.78784158032780126020202925269, −5.18883097760606637189872523705, −4.18179882895477973525290033763, −2.49068836073929250677812316594, −1.21398059441445854471587449743, 2.69838837698258575151434111143, 4.23513884036741774611727503242, 5.30308468203983860572976377039, 6.26076104655416249315721672620, 7.947734520789179949784211623038, 8.913840394035508405321525993329, 11.19169073900779693328105679686, 11.71571852892151982396208423620, 13.48090446655789076953423636124, 13.84102175799580738477170275041

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