Properties

Label 2-69-1.1-c7-0-12
Degree $2$
Conductor $69$
Sign $-1$
Analytic cond. $21.5545$
Root an. cond. $4.64268$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.64·2-s − 27·3-s − 34.8·4-s − 306.·5-s + 260.·6-s + 733.·7-s + 1.57e3·8-s + 729·9-s + 2.96e3·10-s + 3.88e3·11-s + 942.·12-s − 5.97e3·13-s − 7.07e3·14-s + 8.28e3·15-s − 1.06e4·16-s + 1.40e4·17-s − 7.03e3·18-s + 1.10e4·19-s + 1.07e4·20-s − 1.97e4·21-s − 3.74e4·22-s + 1.21e4·23-s − 4.24e4·24-s + 1.60e4·25-s + 5.76e4·26-s − 1.96e4·27-s − 2.55e4·28-s + ⋯
L(s)  = 1  − 0.852·2-s − 0.577·3-s − 0.272·4-s − 1.09·5-s + 0.492·6-s + 0.807·7-s + 1.08·8-s + 0.333·9-s + 0.936·10-s + 0.879·11-s + 0.157·12-s − 0.754·13-s − 0.688·14-s + 0.634·15-s − 0.653·16-s + 0.692·17-s − 0.284·18-s + 0.368·19-s + 0.299·20-s − 0.466·21-s − 0.750·22-s + 0.208·23-s − 0.626·24-s + 0.205·25-s + 0.643·26-s − 0.192·27-s − 0.220·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-1$
Analytic conductor: \(21.5545\)
Root analytic conductor: \(4.64268\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 69,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 27T \)
23 \( 1 - 1.21e4T \)
good2 \( 1 + 9.64T + 128T^{2} \)
5 \( 1 + 306.T + 7.81e4T^{2} \)
7 \( 1 - 733.T + 8.23e5T^{2} \)
11 \( 1 - 3.88e3T + 1.94e7T^{2} \)
13 \( 1 + 5.97e3T + 6.27e7T^{2} \)
17 \( 1 - 1.40e4T + 4.10e8T^{2} \)
19 \( 1 - 1.10e4T + 8.93e8T^{2} \)
29 \( 1 - 116.T + 1.72e10T^{2} \)
31 \( 1 + 6.55e4T + 2.75e10T^{2} \)
37 \( 1 + 3.82e5T + 9.49e10T^{2} \)
41 \( 1 + 3.70e5T + 1.94e11T^{2} \)
43 \( 1 - 3.68e5T + 2.71e11T^{2} \)
47 \( 1 - 8.30e5T + 5.06e11T^{2} \)
53 \( 1 + 5.45e5T + 1.17e12T^{2} \)
59 \( 1 - 1.87e6T + 2.48e12T^{2} \)
61 \( 1 + 1.20e6T + 3.14e12T^{2} \)
67 \( 1 + 4.04e6T + 6.06e12T^{2} \)
71 \( 1 - 2.77e6T + 9.09e12T^{2} \)
73 \( 1 + 2.58e6T + 1.10e13T^{2} \)
79 \( 1 - 1.28e6T + 1.92e13T^{2} \)
83 \( 1 - 5.20e6T + 2.71e13T^{2} \)
89 \( 1 + 2.72e6T + 4.42e13T^{2} \)
97 \( 1 + 7.55e6T + 8.07e13T^{2} \)
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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

−12.31687977693731512091494175613, −11.54437629033133448957903266295, −10.46115552017304227796139020871, −9.209179108719519092184621265339, −8.017565846059859975597720437858, −7.17458608379544543255410979668, −5.12402218594518555340243433072, −3.97028470065954718883757183443, −1.31703756846590892476662074952, 0, 1.31703756846590892476662074952, 3.97028470065954718883757183443, 5.12402218594518555340243433072, 7.17458608379544543255410979668, 8.017565846059859975597720437858, 9.209179108719519092184621265339, 10.46115552017304227796139020871, 11.54437629033133448957903266295, 12.31687977693731512091494175613

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