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Properties

Label 2-69-69.68-c7-0-16
Degree $2$
Conductor $69$
Sign $0.558 + 0.829i$
Analytic cond. $21.5545$
Root an. cond. $4.64268$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 20.7i·2-s + (11.8 + 45.2i)3-s − 301.·4-s − 470.·5-s + (937. − 244. i)6-s − 481. i·7-s + 3.58e3i·8-s + (−1.90e3 + 1.06e3i)9-s + 9.75e3i·10-s + 6.88e3·11-s + (−3.55e3 − 1.36e4i)12-s − 1.06e4·13-s − 9.97e3·14-s + (−5.56e3 − 2.13e4i)15-s + 3.57e4·16-s + 2.16e4·17-s + ⋯
L(s)  = 1  − 1.83i·2-s + (0.252 + 0.967i)3-s − 2.35·4-s − 1.68·5-s + (1.77 − 0.462i)6-s − 0.530i·7-s + 2.47i·8-s + (−0.872 + 0.489i)9-s + 3.08i·10-s + 1.55·11-s + (−0.594 − 2.27i)12-s − 1.34·13-s − 0.971·14-s + (−0.425 − 1.63i)15-s + 2.18·16-s + 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.558 + 0.829i$
Analytic conductor: \(21.5545\)
Root analytic conductor: \(4.64268\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :7/2),\ 0.558 + 0.829i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.908442 - 0.483758i\)
\(L(\frac12)\) \(\approx\) \(0.908442 - 0.483758i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-11.8 - 45.2i)T \)
23 \( 1 + (-5.50e4 + 1.92e4i)T \)
good2 \( 1 + 20.7iT - 128T^{2} \)
5 \( 1 + 470.T + 7.81e4T^{2} \)
7 \( 1 + 481. iT - 8.23e5T^{2} \)
11 \( 1 - 6.88e3T + 1.94e7T^{2} \)
13 \( 1 + 1.06e4T + 6.27e7T^{2} \)
17 \( 1 - 2.16e4T + 4.10e8T^{2} \)
19 \( 1 - 7.53e3iT - 8.93e8T^{2} \)
29 \( 1 + 5.28e4iT - 1.72e10T^{2} \)
31 \( 1 - 2.19e4T + 2.75e10T^{2} \)
37 \( 1 + 2.92e5iT - 9.49e10T^{2} \)
41 \( 1 - 1.29e5iT - 1.94e11T^{2} \)
43 \( 1 - 9.01e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.13e6iT - 5.06e11T^{2} \)
53 \( 1 - 7.97e4T + 1.17e12T^{2} \)
59 \( 1 - 4.80e5iT - 2.48e12T^{2} \)
61 \( 1 - 2.34e5iT - 3.14e12T^{2} \)
67 \( 1 - 8.12e5iT - 6.06e12T^{2} \)
71 \( 1 - 3.86e6iT - 9.09e12T^{2} \)
73 \( 1 - 2.91e6T + 1.10e13T^{2} \)
79 \( 1 + 7.24e6iT - 1.92e13T^{2} \)
83 \( 1 - 4.03e6T + 2.71e13T^{2} \)
89 \( 1 - 1.11e6T + 4.42e13T^{2} \)
97 \( 1 + 3.36e5iT - 8.07e13T^{2} \)
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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

−12.53026081733960288508344485795, −11.75653566073955747993628061707, −11.06103059604908143639041092939, −9.931248151298874540859849653824, −9.011683375065215894833019616194, −7.70318653367391241603506990248, −4.64181595747677498462226428579, −3.95899272334128774708131477839, −3.01250410675300643785295656579, −0.817203200980177367653053343734, 0.57190614013687425658587013352, 3.63045622425667996449160392818, 5.17385820822684981732219843788, 6.78312560051167973490792179801, 7.35304258523641825204024786235, 8.339882150404099333157565306627, 9.207364273404777433197690540068, 11.84182078933925180588273974524, 12.37695060408331393583689377055, 13.93232995085629081323493510376

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