Properties

Label 2-69-69.68-c5-0-36
Degree $2$
Conductor $69$
Sign $0.656 - 0.754i$
Analytic cond. $11.0664$
Root an. cond. $3.32663$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.59i·2-s + (−8.65 − 12.9i)3-s − 60.0·4-s − 42.3·5-s + (−124. + 83.0i)6-s − 238. i·7-s + 269. i·8-s + (−93.1 + 224. i)9-s + 406. i·10-s + 261.·11-s + (519. + 778. i)12-s − 36.2·13-s − 2.29e3·14-s + (366. + 548. i)15-s + 662.·16-s + 1.88e3·17-s + ⋯
L(s)  = 1  − 1.69i·2-s + (−0.555 − 0.831i)3-s − 1.87·4-s − 0.757·5-s + (−1.41 + 0.941i)6-s − 1.84i·7-s + 1.48i·8-s + (−0.383 + 0.923i)9-s + 1.28i·10-s + 0.651·11-s + (1.04 + 1.56i)12-s − 0.0594·13-s − 3.12·14-s + (0.420 + 0.629i)15-s + 0.646·16-s + 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(11.0664\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ 0.656 - 0.754i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.610914 + 0.278066i\)
\(L(\frac12)\) \(\approx\) \(0.610914 + 0.278066i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.65 + 12.9i)T \)
23 \( 1 + (666. + 2.44e3i)T \)
good2 \( 1 + 9.59iT - 32T^{2} \)
5 \( 1 + 42.3T + 3.12e3T^{2} \)
7 \( 1 + 238. iT - 1.68e4T^{2} \)
11 \( 1 - 261.T + 1.61e5T^{2} \)
13 \( 1 + 36.2T + 3.71e5T^{2} \)
17 \( 1 - 1.88e3T + 1.41e6T^{2} \)
19 \( 1 + 1.30e3iT - 2.47e6T^{2} \)
29 \( 1 - 2.23e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.17e3T + 2.86e7T^{2} \)
37 \( 1 - 1.19e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.39e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.15e4iT - 1.47e8T^{2} \)
47 \( 1 + 104. iT - 2.29e8T^{2} \)
53 \( 1 - 3.04e3T + 4.18e8T^{2} \)
59 \( 1 + 4.22e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.96e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.31e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.48e3iT - 1.80e9T^{2} \)
73 \( 1 + 3.49e3T + 2.07e9T^{2} \)
79 \( 1 - 2.16e4iT - 3.07e9T^{2} \)
83 \( 1 + 4.03e4T + 3.93e9T^{2} \)
89 \( 1 - 2.50e4T + 5.58e9T^{2} \)
97 \( 1 + 8.61e4iT - 8.58e9T^{2} \)
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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

−12.50283974690728581228964773566, −11.63783008117706871733315652128, −10.84467936826779633992157476269, −9.923158206695158653246763414648, −8.068285149987747937091137054087, −6.90788236748419907742820843593, −4.60386370232914255091336355898, −3.40824395572123513522601390336, −1.31522861588620541541587232203, −0.37237203189763091650321085181, 3.82985857043809944806834221220, 5.45752520983824422133977222220, 5.93700046501717232252050256817, 7.64440352657380745444959328109, 8.787531288712149304957294510419, 9.660545463767386945374119886751, 11.63543700270146604344015392360, 12.33956974564281260535577156679, 14.39818211507297361470788134427, 14.97572135366819290871108781245

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