Properties

Label 2-69-1.1-c7-0-10
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  − 17.8·2-s − 27·3-s + 189.·4-s − 31.0·5-s + 481.·6-s − 1.24e3·7-s − 1.10e3·8-s + 729·9-s + 553.·10-s − 939.·11-s − 5.12e3·12-s + 1.01e4·13-s + 2.22e4·14-s + 838.·15-s − 4.66e3·16-s + 3.39e4·17-s − 1.29e4·18-s + 1.87e4·19-s − 5.89e3·20-s + 3.36e4·21-s + 1.67e4·22-s + 1.21e4·23-s + 2.97e4·24-s − 7.71e4·25-s − 1.80e5·26-s − 1.96e4·27-s − 2.36e5·28-s + ⋯
L(s)  = 1  − 1.57·2-s − 0.577·3-s + 1.48·4-s − 0.111·5-s + 0.909·6-s − 1.37·7-s − 0.760·8-s + 0.333·9-s + 0.174·10-s − 0.212·11-s − 0.855·12-s + 1.28·13-s + 2.16·14-s + 0.0641·15-s − 0.284·16-s + 1.67·17-s − 0.525·18-s + 0.626·19-s − 0.164·20-s + 0.793·21-s + 0.335·22-s + 0.208·23-s + 0.438·24-s − 0.987·25-s − 2.01·26-s − 0.192·27-s − 2.03·28-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 \( 1 + 17.8T + 128T^{2} \)
5 \( 1 + 31.0T + 7.81e4T^{2} \)
7 \( 1 + 1.24e3T + 8.23e5T^{2} \)
11 \( 1 + 939.T + 1.94e7T^{2} \)
13 \( 1 - 1.01e4T + 6.27e7T^{2} \)
17 \( 1 - 3.39e4T + 4.10e8T^{2} \)
19 \( 1 - 1.87e4T + 8.93e8T^{2} \)
29 \( 1 - 9.17e4T + 1.72e10T^{2} \)
31 \( 1 + 1.87e5T + 2.75e10T^{2} \)
37 \( 1 + 1.12e5T + 9.49e10T^{2} \)
41 \( 1 - 2.18e5T + 1.94e11T^{2} \)
43 \( 1 + 9.10e5T + 2.71e11T^{2} \)
47 \( 1 - 5.40e4T + 5.06e11T^{2} \)
53 \( 1 + 1.35e5T + 1.17e12T^{2} \)
59 \( 1 + 2.26e6T + 2.48e12T^{2} \)
61 \( 1 - 1.24e5T + 3.14e12T^{2} \)
67 \( 1 - 3.30e6T + 6.06e12T^{2} \)
71 \( 1 + 7.40e5T + 9.09e12T^{2} \)
73 \( 1 - 2.68e6T + 1.10e13T^{2} \)
79 \( 1 + 6.55e6T + 1.92e13T^{2} \)
83 \( 1 - 3.21e6T + 2.71e13T^{2} \)
89 \( 1 + 1.14e7T + 4.42e13T^{2} \)
97 \( 1 + 1.41e7T + 8.07e13T^{2} \)
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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

−12.43183344167925362406809073953, −11.24770520831105459157346927361, −10.16034362324882423917146374502, −9.486078391538998439272073801803, −8.152060366775460414518200455858, −6.96322446520949471938927925207, −5.80495079534187389541126855548, −3.38695247592134488151006307058, −1.24661487730044404514098620275, 0, 1.24661487730044404514098620275, 3.38695247592134488151006307058, 5.80495079534187389541126855548, 6.96322446520949471938927925207, 8.152060366775460414518200455858, 9.486078391538998439272073801803, 10.16034362324882423917146374502, 11.24770520831105459157346927361, 12.43183344167925362406809073953

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