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Properties

Label 2-69-3.2-c6-0-38
Degree $2$
Conductor $69$
Sign $-0.751 + 0.659i$
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  − 5.20i·2-s + (17.8 + 20.2i)3-s + 36.8·4-s − 125. i·5-s + (105. − 92.7i)6-s − 678.·7-s − 525. i·8-s + (−94.5 + 722. i)9-s − 655.·10-s − 1.92e3i·11-s + (656. + 748. i)12-s − 1.92e3·13-s + 3.53e3i·14-s + (2.55e3 − 2.24e3i)15-s − 375.·16-s − 4.42e3i·17-s + ⋯
L(s)  = 1  − 0.650i·2-s + (0.659 + 0.751i)3-s + 0.576·4-s − 1.00i·5-s + (0.489 − 0.429i)6-s − 1.97·7-s − 1.02i·8-s + (−0.129 + 0.991i)9-s − 0.655·10-s − 1.44i·11-s + (0.380 + 0.433i)12-s − 0.877·13-s + 1.28i·14-s + (0.756 − 0.663i)15-s − 0.0915·16-s − 0.900i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.751 + 0.659i$
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: \((2,\ 69,\ (\ :3),\ -0.751 + 0.659i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.547516 - 1.45383i\)
\(L(\frac12)\) \(\approx\) \(0.547516 - 1.45383i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-17.8 - 20.2i)T \)
23 \( 1 + 2.53e3iT \)
good2 \( 1 + 5.20iT - 64T^{2} \)
5 \( 1 + 125. iT - 1.56e4T^{2} \)
7 \( 1 + 678.T + 1.17e5T^{2} \)
11 \( 1 + 1.92e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.92e3T + 4.82e6T^{2} \)
17 \( 1 + 4.42e3iT - 2.41e7T^{2} \)
19 \( 1 - 6.71e3T + 4.70e7T^{2} \)
29 \( 1 - 1.19e4iT - 5.94e8T^{2} \)
31 \( 1 + 2.32e4T + 8.87e8T^{2} \)
37 \( 1 + 6.84e4T + 2.56e9T^{2} \)
41 \( 1 + 3.83e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.19e5T + 6.32e9T^{2} \)
47 \( 1 + 3.81e4iT - 1.07e10T^{2} \)
53 \( 1 + 3.98e4iT - 2.21e10T^{2} \)
59 \( 1 - 4.57e4iT - 4.21e10T^{2} \)
61 \( 1 - 1.43e5T + 5.15e10T^{2} \)
67 \( 1 + 3.13e5T + 9.04e10T^{2} \)
71 \( 1 - 2.14e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.14e4T + 1.51e11T^{2} \)
79 \( 1 - 5.15e5T + 2.43e11T^{2} \)
83 \( 1 - 5.54e4iT - 3.26e11T^{2} \)
89 \( 1 - 4.19e5iT - 4.96e11T^{2} \)
97 \( 1 + 4.24e5T + 8.32e11T^{2} \)
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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

−12.98237367388017923899042149452, −12.06949569378840501567612532928, −10.65207592962422367499895380697, −9.614853117182380657476511644337, −8.956278973308925155531583933974, −7.15386770931485501987003418999, −5.50968859014177066140893046245, −3.59766560377406838516455373325, −2.77729433025496128745664963747, −0.51274712692482512764593093216, 2.26668235199781624741237733176, 3.29858377999344194598619625934, 6.08131469316599658620326335294, 7.00779274215919292095688553560, 7.44639303234361308680164506836, 9.359949411233936889831850343920, 10.31995269742948827059842750858, 12.09108212930100935271604847299, 12.83028423271033651471273144459, 14.17886537904803690958171187086

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