Properties

Label 2-69-1.1-c5-0-12
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  + 9.27·2-s + 9·3-s + 53.9·4-s − 37.4·5-s + 83.4·6-s + 154.·7-s + 203.·8-s + 81·9-s − 347.·10-s + 521.·11-s + 485.·12-s − 28.8·13-s + 1.43e3·14-s − 337.·15-s + 163.·16-s − 428.·17-s + 751.·18-s − 2.56e3·19-s − 2.02e3·20-s + 1.39e3·21-s + 4.83e3·22-s − 529·23-s + 1.83e3·24-s − 1.71e3·25-s − 267.·26-s + 729·27-s + 8.36e3·28-s + ⋯
L(s)  = 1  + 1.63·2-s + 0.577·3-s + 1.68·4-s − 0.670·5-s + 0.946·6-s + 1.19·7-s + 1.12·8-s + 0.333·9-s − 1.09·10-s + 1.29·11-s + 0.974·12-s − 0.0473·13-s + 1.95·14-s − 0.387·15-s + 0.159·16-s − 0.359·17-s + 0.546·18-s − 1.62·19-s − 1.13·20-s + 0.689·21-s + 2.12·22-s − 0.208·23-s + 0.650·24-s − 0.550·25-s − 0.0776·26-s + 0.192·27-s + 2.01·28-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

\(L(3)\) \(\approx\) \(5.006866200\)
\(L(\frac12)\) \(\approx\) \(5.006866200\)
\(L(\frac{7}{2})\) 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 \( 1 - 9.27T + 32T^{2} \)
5 \( 1 + 37.4T + 3.12e3T^{2} \)
7 \( 1 - 154.T + 1.68e4T^{2} \)
11 \( 1 - 521.T + 1.61e5T^{2} \)
13 \( 1 + 28.8T + 3.71e5T^{2} \)
17 \( 1 + 428.T + 1.41e6T^{2} \)
19 \( 1 + 2.56e3T + 2.47e6T^{2} \)
29 \( 1 + 2.40e3T + 2.05e7T^{2} \)
31 \( 1 + 2.13e3T + 2.86e7T^{2} \)
37 \( 1 - 3.65e3T + 6.93e7T^{2} \)
41 \( 1 - 1.61e4T + 1.15e8T^{2} \)
43 \( 1 - 6.50e3T + 1.47e8T^{2} \)
47 \( 1 + 2.07e4T + 2.29e8T^{2} \)
53 \( 1 - 3.43e4T + 4.18e8T^{2} \)
59 \( 1 + 1.82e4T + 7.14e8T^{2} \)
61 \( 1 + 2.65e4T + 8.44e8T^{2} \)
67 \( 1 + 2.63e4T + 1.35e9T^{2} \)
71 \( 1 - 3.94e4T + 1.80e9T^{2} \)
73 \( 1 + 3.74e4T + 2.07e9T^{2} \)
79 \( 1 - 3.73e4T + 3.07e9T^{2} \)
83 \( 1 - 7.69e4T + 3.93e9T^{2} \)
89 \( 1 - 6.29e4T + 5.58e9T^{2} \)
97 \( 1 - 6.14e4T + 8.58e9T^{2} \)
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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

−13.94086467503151276917945234364, −12.78015199032099192306601262343, −11.77254902314266058723494136232, −11.00021693884384860269396909067, −8.919235447076406825886199462614, −7.61793210200202970135012910545, −6.24162901151056270447519969965, −4.54929984042531072256859165199, −3.84756711507265125489847511679, −2.02406169764490789053824577434, 2.02406169764490789053824577434, 3.84756711507265125489847511679, 4.54929984042531072256859165199, 6.24162901151056270447519969965, 7.61793210200202970135012910545, 8.919235447076406825886199462614, 11.00021693884384860269396909067, 11.77254902314266058723494136232, 12.78015199032099192306601262343, 13.94086467503151276917945234364

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