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Properties

Label 2-69-3.2-c2-0-4
Degree $2$
Conductor $69$
Sign $0.981 - 0.190i$
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  − 1.90i·2-s + (0.570 + 2.94i)3-s + 0.360·4-s + 9.03i·5-s + (5.61 − 1.08i)6-s + 1.87·7-s − 8.31i·8-s + (−8.34 + 3.36i)9-s + 17.2·10-s − 15.0i·11-s + (0.206 + 1.06i)12-s + 4.90·13-s − 3.58i·14-s + (−26.6 + 5.16i)15-s − 14.4·16-s − 10.8i·17-s + ⋯
L(s)  = 1  − 0.953i·2-s + (0.190 + 0.981i)3-s + 0.0902·4-s + 1.80i·5-s + (0.936 − 0.181i)6-s + 0.268·7-s − 1.03i·8-s + (−0.927 + 0.373i)9-s + 1.72·10-s − 1.36i·11-s + (0.0171 + 0.0885i)12-s + 0.377·13-s − 0.256i·14-s + (−1.77 + 0.344i)15-s − 0.901·16-s − 0.635i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.981 - 0.190i$
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: \((2,\ 69,\ (\ :1),\ 0.981 - 0.190i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.40595 + 0.135010i\)
\(L(\frac12)\) \(\approx\) \(1.40595 + 0.135010i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.570 - 2.94i)T \)
23 \( 1 - 4.79iT \)
good2 \( 1 + 1.90iT - 4T^{2} \)
5 \( 1 - 9.03iT - 25T^{2} \)
7 \( 1 - 1.87T + 49T^{2} \)
11 \( 1 + 15.0iT - 121T^{2} \)
13 \( 1 - 4.90T + 169T^{2} \)
17 \( 1 + 10.8iT - 289T^{2} \)
19 \( 1 - 20.4T + 361T^{2} \)
29 \( 1 - 18.6iT - 841T^{2} \)
31 \( 1 + 20.8T + 961T^{2} \)
37 \( 1 - 18.5T + 1.36e3T^{2} \)
41 \( 1 + 71.5iT - 1.68e3T^{2} \)
43 \( 1 - 37.4T + 1.84e3T^{2} \)
47 \( 1 - 47.3iT - 2.20e3T^{2} \)
53 \( 1 + 25.0iT - 2.80e3T^{2} \)
59 \( 1 - 88.7iT - 3.48e3T^{2} \)
61 \( 1 + 86.1T + 3.72e3T^{2} \)
67 \( 1 + 18.0T + 4.48e3T^{2} \)
71 \( 1 - 28.8iT - 5.04e3T^{2} \)
73 \( 1 + 27.2T + 5.32e3T^{2} \)
79 \( 1 + 9.77T + 6.24e3T^{2} \)
83 \( 1 + 42.3iT - 6.88e3T^{2} \)
89 \( 1 - 106. iT - 7.92e3T^{2} \)
97 \( 1 - 76.9T + 9.40e3T^{2} \)
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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

−14.41288222391012536694826461578, −13.71112339200877566619039454697, −11.65944895447273341595167492746, −11.03206373845797759542531078714, −10.45937468536067400301776683863, −9.254908876317654845844327521937, −7.42911597870299130037435403118, −5.92675129945681798061111119847, −3.59239744971554407590353622452, −2.82978296591482075591427947792, 1.66748491692940475399602463307, 4.83177356880290715636907029697, 6.03975836565307811451158655421, 7.51922770826941352827788469477, 8.248043401823434485924094947742, 9.371625087899312772996306447680, 11.57478906148683687459018550424, 12.47094236299250666350189404935, 13.32162133575118879299983702684, 14.54596403236598764421189002048

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