Properties

Label 2-69-1.1-c5-0-1
Degree $2$
Conductor $69$
Sign $1$
Analytic cond. $11.0664$
Root an. cond. $3.32663$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.7·2-s − 9·3-s + 83.9·4-s + 41.6·5-s + 96.9·6-s + 0.236·7-s − 560.·8-s + 81·9-s − 448.·10-s − 421.·11-s − 755.·12-s + 254.·13-s − 2.55·14-s − 374.·15-s + 3.34e3·16-s − 975.·17-s − 872.·18-s + 2.03e3·19-s + 3.49e3·20-s − 2.13·21-s + 4.54e3·22-s + 529·23-s + 5.04e3·24-s − 1.39e3·25-s − 2.74e3·26-s − 729·27-s + 19.8·28-s + ⋯
L(s)  = 1  − 1.90·2-s − 0.577·3-s + 2.62·4-s + 0.744·5-s + 1.09·6-s + 0.00182·7-s − 3.09·8-s + 0.333·9-s − 1.41·10-s − 1.05·11-s − 1.51·12-s + 0.417·13-s − 0.00347·14-s − 0.429·15-s + 3.26·16-s − 0.818·17-s − 0.634·18-s + 1.29·19-s + 1.95·20-s − 0.00105·21-s + 2.00·22-s + 0.208·23-s + 1.78·24-s − 0.445·25-s − 0.795·26-s − 0.192·27-s + 0.00479·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $1$
Analytic conductor: \(11.0664\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6232221173\)
\(L(\frac12)\) \(\approx\) \(0.6232221173\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
23 \( 1 - 529T \)
good2 \( 1 + 10.7T + 32T^{2} \)
5 \( 1 - 41.6T + 3.12e3T^{2} \)
7 \( 1 - 0.236T + 1.68e4T^{2} \)
11 \( 1 + 421.T + 1.61e5T^{2} \)
13 \( 1 - 254.T + 3.71e5T^{2} \)
17 \( 1 + 975.T + 1.41e6T^{2} \)
19 \( 1 - 2.03e3T + 2.47e6T^{2} \)
29 \( 1 - 2.67e3T + 2.05e7T^{2} \)
31 \( 1 - 9.03e3T + 2.86e7T^{2} \)
37 \( 1 + 1.26e4T + 6.93e7T^{2} \)
41 \( 1 - 1.01e4T + 1.15e8T^{2} \)
43 \( 1 - 1.95e4T + 1.47e8T^{2} \)
47 \( 1 - 2.76e4T + 2.29e8T^{2} \)
53 \( 1 - 1.08e4T + 4.18e8T^{2} \)
59 \( 1 - 1.19e4T + 7.14e8T^{2} \)
61 \( 1 - 3.98e4T + 8.44e8T^{2} \)
67 \( 1 + 2.85e4T + 1.35e9T^{2} \)
71 \( 1 - 5.21e4T + 1.80e9T^{2} \)
73 \( 1 - 5.69e4T + 2.07e9T^{2} \)
79 \( 1 - 2.31e4T + 3.07e9T^{2} \)
83 \( 1 + 1.83e4T + 3.93e9T^{2} \)
89 \( 1 - 4.73e4T + 5.58e9T^{2} \)
97 \( 1 + 1.40e5T + 8.58e9T^{2} \)
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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

−13.69217455460158908780875074479, −12.16031416838799569990171706179, −11.01103954147332120567672158061, −10.23181548528104791341494476831, −9.299326725517822566666347237099, −8.056400359726645824371823969231, −6.85340485695555742582358487104, −5.64430241437460740915080945928, −2.43766725803651935698219596128, −0.842382676172294653687089750060, 0.842382676172294653687089750060, 2.43766725803651935698219596128, 5.64430241437460740915080945928, 6.85340485695555742582358487104, 8.056400359726645824371823969231, 9.299326725517822566666347237099, 10.23181548528104791341494476831, 11.01103954147332120567672158061, 12.16031416838799569990171706179, 13.69217455460158908780875074479

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