Properties

Label 2-69-69.68-c3-0-4
Degree $2$
Conductor $69$
Sign $-0.922 + 0.384i$
Analytic cond. $4.07113$
Root an. cond. $2.01770$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.79i·2-s + (2 + 4.79i)3-s − 14.9·4-s + (−22.9 + 9.59i)6-s − 33.5i·8-s + (−18.9 + 19.1i)9-s + (−29.9 − 71.9i)12-s + 74·13-s + 40.9·16-s + (−91.9 − 91.1i)18-s + 110. i·23-s + (160. − 67.1i)24-s − 125·25-s + 354. i·26-s + (−129. − 52.7i)27-s + ⋯
L(s)  = 1  + 1.69i·2-s + (0.384 + 0.922i)3-s − 1.87·4-s + (−1.56 + 0.652i)6-s − 1.48i·8-s + (−0.703 + 0.710i)9-s + (−0.721 − 1.73i)12-s + 1.57·13-s + 0.640·16-s + (−1.20 − 1.19i)18-s + 0.999i·23-s + (1.36 − 0.571i)24-s − 25-s + 2.67i·26-s + (−0.926 − 0.376i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.922 + 0.384i$
Analytic conductor: \(4.07113\)
Root analytic conductor: \(2.01770\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3/2),\ -0.922 + 0.384i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.275940 - 1.37859i\)
\(L(\frac12)\) \(\approx\) \(0.275940 - 1.37859i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2 - 4.79i)T \)
23 \( 1 - 110. iT \)
good2 \( 1 - 4.79iT - 8T^{2} \)
5 \( 1 + 125T^{2} \)
7 \( 1 - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 74T + 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
29 \( 1 - 134. iT - 2.43e4T^{2} \)
31 \( 1 - 344T + 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 - 306. iT - 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 642. iT - 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + 815. iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 - 220. iT - 3.57e5T^{2} \)
73 \( 1 - 1.22e3T + 3.89e5T^{2} \)
79 \( 1 - 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−15.27421098183021062462505979863, −14.00130746800366898984595644535, −13.45471690948708818404110321292, −11.35819148611223138738775427391, −9.900023497948015880628023172706, −8.748361864258654657959388936913, −7.967895682497640834627472920994, −6.38010374506982623737782445471, −5.22147457768017450620088117595, −3.80305279376031239900559755841, 0.996139940795100827595483904932, 2.53801662311772924551648335151, 3.98297866686422702784859810415, 6.23143845428292983270152484031, 8.134491939867327801130737487398, 9.134582039500908511314170686275, 10.48819360561422095006563385658, 11.56317134080674220664646831262, 12.37440759397331378666586814780, 13.44994154409642373551389140765

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