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Properties

Label 2-69-3.2-c2-0-5
Degree $2$
Conductor $69$
Sign $-0.230 - 0.973i$
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  + 2.92i·2-s + (2.91 − 0.690i)3-s − 4.56·4-s + 2.77i·5-s + (2.01 + 8.54i)6-s − 1.17·7-s − 1.65i·8-s + (8.04 − 4.02i)9-s − 8.12·10-s − 8.93i·11-s + (−13.3 + 3.15i)12-s − 10.2·13-s − 3.43i·14-s + (1.91 + 8.10i)15-s − 13.4·16-s − 21.1i·17-s + ⋯
L(s)  = 1  + 1.46i·2-s + (0.973 − 0.230i)3-s − 1.14·4-s + 0.554i·5-s + (0.336 + 1.42i)6-s − 0.167·7-s − 0.207i·8-s + (0.894 − 0.447i)9-s − 0.812·10-s − 0.812i·11-s + (−1.11 + 0.262i)12-s − 0.792·13-s − 0.245i·14-s + (0.127 + 0.540i)15-s − 0.838·16-s − 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.230 - 0.973i$
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: \((2,\ 69,\ (\ :1),\ -0.230 - 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.959841 + 1.21317i\)
\(L(\frac12)\) \(\approx\) \(0.959841 + 1.21317i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.91 + 0.690i)T \)
23 \( 1 - 4.79iT \)
good2 \( 1 - 2.92iT - 4T^{2} \)
5 \( 1 - 2.77iT - 25T^{2} \)
7 \( 1 + 1.17T + 49T^{2} \)
11 \( 1 + 8.93iT - 121T^{2} \)
13 \( 1 + 10.2T + 169T^{2} \)
17 \( 1 + 21.1iT - 289T^{2} \)
19 \( 1 - 27.2T + 361T^{2} \)
29 \( 1 - 9.80iT - 841T^{2} \)
31 \( 1 + 32.5T + 961T^{2} \)
37 \( 1 + 46.6T + 1.36e3T^{2} \)
41 \( 1 - 59.6iT - 1.68e3T^{2} \)
43 \( 1 + 10.3T + 1.84e3T^{2} \)
47 \( 1 + 21.5iT - 2.20e3T^{2} \)
53 \( 1 + 75.1iT - 2.80e3T^{2} \)
59 \( 1 - 65.9iT - 3.48e3T^{2} \)
61 \( 1 - 29.5T + 3.72e3T^{2} \)
67 \( 1 + 38.5T + 4.48e3T^{2} \)
71 \( 1 - 138. iT - 5.04e3T^{2} \)
73 \( 1 + 73.4T + 5.32e3T^{2} \)
79 \( 1 - 28.2T + 6.24e3T^{2} \)
83 \( 1 - 119. iT - 6.88e3T^{2} \)
89 \( 1 + 115. iT - 7.92e3T^{2} \)
97 \( 1 - 116.T + 9.40e3T^{2} \)
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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

−14.67860556998521422710257767873, −14.17762199346039110355059701457, −13.23577442984432423343559246188, −11.56909229294207291180839077323, −9.773374687076915641109511683930, −8.714360911588809737210284164121, −7.49134499783614463633928822423, −6.84210594780368339977879343216, −5.20198499628981668077665026489, −3.06799659296372391278382621368, 1.84070447263795107187432695738, 3.40695198071030186814313196283, 4.74946863839051563646004565652, 7.35780623824651173570287822756, 8.894093040343067714938785578037, 9.742370780761987901685200998742, 10.63268619637330299007680712647, 12.22795115852396032860795846861, 12.76300068230181012274791540244, 13.89060274212098329238187692418

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