Properties

Label 2-69-3.2-c2-0-12
Degree $2$
Conductor $69$
Sign $-0.987 + 0.159i$
Analytic cond. $1.88011$
Root an. cond. $1.37117$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.07i·2-s + (−0.479 − 2.96i)3-s − 5.48·4-s + 0.529i·5-s + (−9.12 + 1.47i)6-s + 5.42·7-s + 4.57i·8-s + (−8.53 + 2.84i)9-s + 1.62·10-s + 6.59i·11-s + (2.63 + 16.2i)12-s + 12.8·13-s − 16.7i·14-s + (1.56 − 0.253i)15-s − 7.85·16-s − 30.3i·17-s + ⋯
L(s)  = 1  − 1.53i·2-s + (−0.159 − 0.987i)3-s − 1.37·4-s + 0.105i·5-s + (−1.52 + 0.246i)6-s + 0.775·7-s + 0.571i·8-s + (−0.948 + 0.315i)9-s + 0.162·10-s + 0.599i·11-s + (0.219 + 1.35i)12-s + 0.986·13-s − 1.19i·14-s + (0.104 − 0.0169i)15-s − 0.490·16-s − 1.78i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.987 + 0.159i$
Analytic conductor: \(1.88011\)
Root analytic conductor: \(1.37117\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1),\ -0.987 + 0.159i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0927233 - 1.15172i\)
\(L(\frac12)\) \(\approx\) \(0.0927233 - 1.15172i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.479 + 2.96i)T \)
23 \( 1 - 4.79iT \)
good2 \( 1 + 3.07iT - 4T^{2} \)
5 \( 1 - 0.529iT - 25T^{2} \)
7 \( 1 - 5.42T + 49T^{2} \)
11 \( 1 - 6.59iT - 121T^{2} \)
13 \( 1 - 12.8T + 169T^{2} \)
17 \( 1 + 30.3iT - 289T^{2} \)
19 \( 1 + 2.33T + 361T^{2} \)
29 \( 1 - 12.8iT - 841T^{2} \)
31 \( 1 - 13.8T + 961T^{2} \)
37 \( 1 - 45.7T + 1.36e3T^{2} \)
41 \( 1 - 60.7iT - 1.68e3T^{2} \)
43 \( 1 + 70.9T + 1.84e3T^{2} \)
47 \( 1 - 66.7iT - 2.20e3T^{2} \)
53 \( 1 + 24.9iT - 2.80e3T^{2} \)
59 \( 1 + 42.0iT - 3.48e3T^{2} \)
61 \( 1 - 94.7T + 3.72e3T^{2} \)
67 \( 1 - 55.1T + 4.48e3T^{2} \)
71 \( 1 - 4.00iT - 5.04e3T^{2} \)
73 \( 1 + 89.8T + 5.32e3T^{2} \)
79 \( 1 + 70.5T + 6.24e3T^{2} \)
83 \( 1 - 43.8iT - 6.88e3T^{2} \)
89 \( 1 - 95.2iT - 7.92e3T^{2} \)
97 \( 1 + 43.0T + 9.40e3T^{2} \)
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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

−13.52683592097723053196115919188, −12.70761498980625210793146990226, −11.53513287319602864771544713103, −11.16956891839942883598152132067, −9.648307984770037745523496341489, −8.294648794281164518148953077630, −6.82467739435992837755236222498, −4.85047524825951537690427452657, −2.82320886301845776404984718018, −1.26518399156166327293133845770, 4.07887140836595090161424937969, 5.41141580656464376513299018192, 6.39123981229901989056821111548, 8.266340733051701276603446499688, 8.703207210906274073163587875072, 10.43927116667867655850391866909, 11.46368591414343550739178960648, 13.31777849054204667527010358843, 14.50175441071321127627058318490, 15.07688881274970928218458498357

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