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Properties

Label 2-69-3.2-c6-0-14
Degree $2$
Conductor $69$
Sign $0.724 - 0.689i$
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  − 8.07i·2-s + (18.6 + 19.5i)3-s − 1.24·4-s + 135. i·5-s + (157. − 150. i)6-s − 41.9·7-s − 506. i·8-s + (−35.8 + 728. i)9-s + 1.09e3·10-s + 1.28e3i·11-s + (−23.1 − 24.3i)12-s − 844.·13-s + 338. i·14-s + (−2.64e3 + 2.51e3i)15-s − 4.17e3·16-s + 8.23e3i·17-s + ⋯
L(s)  = 1  − 1.00i·2-s + (0.689 + 0.724i)3-s − 0.0194·4-s + 1.08i·5-s + (0.731 − 0.696i)6-s − 0.122·7-s − 0.990i·8-s + (−0.0491 + 0.998i)9-s + 1.09·10-s + 0.962i·11-s + (−0.0133 − 0.0140i)12-s − 0.384·13-s + 0.123i·14-s + (−0.784 + 0.746i)15-s − 1.01·16-s + 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.724 - 0.689i$
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: \((2,\ 69,\ (\ :3),\ 0.724 - 0.689i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.14921 + 0.859441i\)
\(L(\frac12)\) \(\approx\) \(2.14921 + 0.859441i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-18.6 - 19.5i)T \)
23 \( 1 - 2.53e3iT \)
good2 \( 1 + 8.07iT - 64T^{2} \)
5 \( 1 - 135. iT - 1.56e4T^{2} \)
7 \( 1 + 41.9T + 1.17e5T^{2} \)
11 \( 1 - 1.28e3iT - 1.77e6T^{2} \)
13 \( 1 + 844.T + 4.82e6T^{2} \)
17 \( 1 - 8.23e3iT - 2.41e7T^{2} \)
19 \( 1 - 826.T + 4.70e7T^{2} \)
29 \( 1 + 1.92e4iT - 5.94e8T^{2} \)
31 \( 1 - 4.40e4T + 8.87e8T^{2} \)
37 \( 1 - 5.00e4T + 2.56e9T^{2} \)
41 \( 1 - 5.71e4iT - 4.75e9T^{2} \)
43 \( 1 - 2.53e4T + 6.32e9T^{2} \)
47 \( 1 + 1.35e5iT - 1.07e10T^{2} \)
53 \( 1 + 6.06e3iT - 2.21e10T^{2} \)
59 \( 1 - 6.56e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.73e5T + 5.15e10T^{2} \)
67 \( 1 + 7.74e4T + 9.04e10T^{2} \)
71 \( 1 + 5.79e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.54e5T + 1.51e11T^{2} \)
79 \( 1 - 5.03e5T + 2.43e11T^{2} \)
83 \( 1 - 1.19e5iT - 3.26e11T^{2} \)
89 \( 1 - 6.99e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.08e6T + 8.32e11T^{2} \)
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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

−13.58762340634150674142664460630, −12.40108588026674607562964883606, −11.13188513980355122190666844640, −10.27234153532430607096706394283, −9.652941069471287307486121403587, −7.898889725124522379841723800356, −6.52570711947047834505877866552, −4.28166760598127532651413920962, −3.08448409868814004025857475528, −2.01950994146193744900798576090, 0.838404025108254340841995695272, 2.74551455535948962809696916084, 4.97074326394943275745868810518, 6.32226234800047968323797000593, 7.50752505379432509190706073744, 8.454892208715896744955256384669, 9.344017995578166530938085679531, 11.44323748644307116734443453431, 12.49666536342400700967967347453, 13.67987697015031666675291335494

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