Properties

Label 2-69-69.68-c1-0-1
Degree $2$
Conductor $69$
Sign $-0.381 - 0.924i$
Analytic cond. $0.550967$
Root an. cond. $0.742272$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.77i·2-s + (1.60 − 0.661i)3-s − 5.72·4-s + (1.83 + 4.44i)6-s − 10.3i·8-s + (2.12 − 2.11i)9-s + (−9.16 + 3.78i)12-s − 2.15·13-s + 17.3·16-s + (5.88 + 5.90i)18-s + 4.79i·23-s + (−6.85 − 16.5i)24-s − 5·25-s − 5.98i·26-s + (2.00 − 4.79i)27-s + ⋯
L(s)  = 1  + 1.96i·2-s + (0.924 − 0.381i)3-s − 2.86·4-s + (0.750 + 1.81i)6-s − 3.66i·8-s + (0.708 − 0.705i)9-s + (−2.64 + 1.09i)12-s − 0.597·13-s + 4.33·16-s + (1.38 + 1.39i)18-s + 0.999i·23-s + (−1.39 − 3.38i)24-s − 25-s − 1.17i·26-s + (0.384 − 0.922i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.381 - 0.924i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ -0.381 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549546 + 0.821742i\)
\(L(\frac12)\) \(\approx\) \(0.549546 + 0.821742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.60 + 0.661i)T \)
23 \( 1 - 4.79iT \)
good2 \( 1 - 2.77iT - 2T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2.15T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
29 \( 1 + 1.58iT - 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 9.52iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 7.14iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 9.59iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 15.0iT - 71T^{2} \)
73 \( 1 - 17.0T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−15.16353665227169549734257720121, −14.20727685589368507064959062056, −13.51719788774222773618275231619, −12.45407780200635261846510504625, −9.816188586384667223400058779979, −8.947640810752518919349471631676, −7.80071370771741418136223586317, −7.06925972431746398824871010416, −5.64190668724476481857015871817, −3.97771951603574081853788893948, 2.22404428139095077576025598609, 3.62212382977332336213232284029, 4.88563067272750609639119986432, 8.023096448523080129214174074782, 9.148366317651305092016703133046, 9.984219849124725706278763249585, 10.93558694291813681729509592016, 12.19539110166688943902157901436, 13.14152482367105755861052683554, 14.06623661750213540591756810991

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