Properties

Label 2-69-3.2-c6-0-4
Degree $2$
Conductor $69$
Sign $0.861 + 0.507i$
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  + 14.0i·2-s + (−13.7 + 23.2i)3-s − 132.·4-s + 135. i·5-s + (−325. − 191. i)6-s − 169.·7-s − 952. i·8-s + (−353. − 637. i)9-s − 1.89e3·10-s + 1.42e3i·11-s + (1.80e3 − 3.07e3i)12-s + 1.38e3·13-s − 2.37e3i·14-s + (−3.14e3 − 1.85e3i)15-s + 4.88e3·16-s + 3.51e3i·17-s + ⋯
L(s)  = 1  + 1.75i·2-s + (−0.507 + 0.861i)3-s − 2.06·4-s + 1.08i·5-s + (−1.50 − 0.888i)6-s − 0.493·7-s − 1.85i·8-s + (−0.484 − 0.874i)9-s − 1.89·10-s + 1.06i·11-s + (1.04 − 1.77i)12-s + 0.629·13-s − 0.864i·14-s + (−0.932 − 0.549i)15-s + 1.19·16-s + 0.714i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.861 + 0.507i$
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: \((2,\ 69,\ (\ :3),\ 0.861 + 0.507i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.712956 - 0.194385i\)
\(L(\frac12)\) \(\approx\) \(0.712956 - 0.194385i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.7 - 23.2i)T \)
23 \( 1 - 2.53e3iT \)
good2 \( 1 - 14.0iT - 64T^{2} \)
5 \( 1 - 135. iT - 1.56e4T^{2} \)
7 \( 1 + 169.T + 1.17e5T^{2} \)
11 \( 1 - 1.42e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.38e3T + 4.82e6T^{2} \)
17 \( 1 - 3.51e3iT - 2.41e7T^{2} \)
19 \( 1 + 695.T + 4.70e7T^{2} \)
29 \( 1 - 4.68e4iT - 5.94e8T^{2} \)
31 \( 1 - 3.86e4T + 8.87e8T^{2} \)
37 \( 1 + 5.83e4T + 2.56e9T^{2} \)
41 \( 1 + 8.86e4iT - 4.75e9T^{2} \)
43 \( 1 - 5.16e4T + 6.32e9T^{2} \)
47 \( 1 + 7.47e3iT - 1.07e10T^{2} \)
53 \( 1 + 2.09e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.48e5iT - 4.21e10T^{2} \)
61 \( 1 + 5.99e4T + 5.15e10T^{2} \)
67 \( 1 + 3.50e5T + 9.04e10T^{2} \)
71 \( 1 - 4.37e5iT - 1.28e11T^{2} \)
73 \( 1 + 8.56e4T + 1.51e11T^{2} \)
79 \( 1 - 6.96e5T + 2.43e11T^{2} \)
83 \( 1 - 5.17e5iT - 3.26e11T^{2} \)
89 \( 1 - 2.86e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.83e5T + 8.32e11T^{2} \)
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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

−14.87515416840083079919972113995, −14.06306891104802090034286203754, −12.54855548405724764893857505727, −10.82162660185940737686509795911, −9.834473153347897028492594717710, −8.622818243881068223057930634015, −7.02390391421777896752250339957, −6.35360807350252956425905612740, −5.06883504478448622654108979311, −3.63885137488645647765680921606, 0.36201909668394874869964687862, 1.20818427209953883965076497065, 2.84928317569868609413577465324, 4.57260170757572372816426383629, 6.04644010091992585640159920409, 8.211913333288177448725081804883, 9.203609825982295217495884228111, 10.57078566545891791418661211631, 11.64605529663994303453881687797, 12.26481748093990341103541093815

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