Properties

Label 2-69-69.68-c7-0-42
Degree $2$
Conductor $69$
Sign $-0.996 + 0.0822i$
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  − 19.2i·2-s + (46.1 + 7.40i)3-s − 242.·4-s + 153.·5-s + (142. − 888. i)6-s − 1.44e3i·7-s + 2.20e3i·8-s + (2.07e3 + 683. i)9-s − 2.95e3i·10-s + 5.06e3·11-s + (−1.11e4 − 1.79e3i)12-s + 9.56e3·13-s − 2.77e4·14-s + (7.10e3 + 1.13e3i)15-s + 1.13e4·16-s − 2.37e4·17-s + ⋯
L(s)  = 1  − 1.70i·2-s + (0.987 + 0.158i)3-s − 1.89·4-s + 0.550·5-s + (0.269 − 1.67i)6-s − 1.59i·7-s + 1.52i·8-s + (0.949 + 0.312i)9-s − 0.935i·10-s + 1.14·11-s + (−1.86 − 0.299i)12-s + 1.20·13-s − 2.70·14-s + (0.543 + 0.0871i)15-s + 0.692·16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0822i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.996 + 0.0822i$
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: \((2,\ 69,\ (\ :7/2),\ -0.996 + 0.0822i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.114086 - 2.76959i\)
\(L(\frac12)\) \(\approx\) \(0.114086 - 2.76959i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-46.1 - 7.40i)T \)
23 \( 1 + (5.66e4 - 1.39e4i)T \)
good2 \( 1 + 19.2iT - 128T^{2} \)
5 \( 1 - 153.T + 7.81e4T^{2} \)
7 \( 1 + 1.44e3iT - 8.23e5T^{2} \)
11 \( 1 - 5.06e3T + 1.94e7T^{2} \)
13 \( 1 - 9.56e3T + 6.27e7T^{2} \)
17 \( 1 + 2.37e4T + 4.10e8T^{2} \)
19 \( 1 + 3.39e4iT - 8.93e8T^{2} \)
29 \( 1 - 1.13e5iT - 1.72e10T^{2} \)
31 \( 1 + 3.66e3T + 2.75e10T^{2} \)
37 \( 1 + 1.86e5iT - 9.49e10T^{2} \)
41 \( 1 + 4.03e5iT - 1.94e11T^{2} \)
43 \( 1 - 2.63e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.77e5iT - 5.06e11T^{2} \)
53 \( 1 - 1.86e6T + 1.17e12T^{2} \)
59 \( 1 - 1.81e5iT - 2.48e12T^{2} \)
61 \( 1 + 2.62e6iT - 3.14e12T^{2} \)
67 \( 1 - 2.20e6iT - 6.06e12T^{2} \)
71 \( 1 - 5.42e6iT - 9.09e12T^{2} \)
73 \( 1 - 5.37e6T + 1.10e13T^{2} \)
79 \( 1 + 4.32e6iT - 1.92e13T^{2} \)
83 \( 1 + 2.97e6T + 2.71e13T^{2} \)
89 \( 1 + 2.86e6T + 4.42e13T^{2} \)
97 \( 1 - 1.05e6iT - 8.07e13T^{2} \)
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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

−12.97716268868820980765683553086, −11.33627660692831945920187467973, −10.53227846446519496258580227269, −9.546025256475456766804193771193, −8.688678536115286647542576176483, −6.89984742441093592524713224824, −4.25546264905588459971435966954, −3.64897359209222710706709425682, −2.01142439721605592705413358875, −0.935919064777645829604242118036, 1.97589826643682789596611645419, 4.05150290621989732271114915436, 5.87075824210108715668679193057, 6.49708862758631182149630632418, 8.171645266601036879113641478822, 8.791370458701207476140056768195, 9.625246512256143867687615002677, 11.95689919454562841428030627377, 13.36721277845264913976636990094, 14.06339817513796104823578728122

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