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Properties

Label 2-69-69.68-c7-0-15
Degree $2$
Conductor $69$
Sign $0.419 - 0.907i$
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  + 20.0i·2-s + (−46.5 + 4.94i)3-s − 272.·4-s − 368.·5-s + (−98.8 − 930. i)6-s + 90.4i·7-s − 2.89e3i·8-s + (2.13e3 − 459. i)9-s − 7.36e3i·10-s − 7.61e3·11-s + (1.26e4 − 1.34e3i)12-s + 684.·13-s − 1.81e3·14-s + (1.71e4 − 1.81e3i)15-s + 2.29e4·16-s − 1.71e4·17-s + ⋯
L(s)  = 1  + 1.76i·2-s + (−0.994 + 0.105i)3-s − 2.12·4-s − 1.31·5-s + (−0.186 − 1.75i)6-s + 0.0996i·7-s − 1.99i·8-s + (0.977 − 0.210i)9-s − 2.32i·10-s − 1.72·11-s + (2.11 − 0.224i)12-s + 0.0863·13-s − 0.176·14-s + (1.30 − 0.139i)15-s + 1.40·16-s − 0.847·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.419 - 0.907i$
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: \((2,\ 69,\ (\ :7/2),\ 0.419 - 0.907i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.139325 + 0.0891052i\)
\(L(\frac12)\) \(\approx\) \(0.139325 + 0.0891052i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (46.5 - 4.94i)T \)
23 \( 1 + (2.99e4 - 5.00e4i)T \)
good2 \( 1 - 20.0iT - 128T^{2} \)
5 \( 1 + 368.T + 7.81e4T^{2} \)
7 \( 1 - 90.4iT - 8.23e5T^{2} \)
11 \( 1 + 7.61e3T + 1.94e7T^{2} \)
13 \( 1 - 684.T + 6.27e7T^{2} \)
17 \( 1 + 1.71e4T + 4.10e8T^{2} \)
19 \( 1 - 4.09e4iT - 8.93e8T^{2} \)
29 \( 1 + 1.61e5iT - 1.72e10T^{2} \)
31 \( 1 - 4.07e4T + 2.75e10T^{2} \)
37 \( 1 - 3.72e5iT - 9.49e10T^{2} \)
41 \( 1 - 4.12e5iT - 1.94e11T^{2} \)
43 \( 1 + 4.34e5iT - 2.71e11T^{2} \)
47 \( 1 - 9.50e5iT - 5.06e11T^{2} \)
53 \( 1 - 3.96e4T + 1.17e12T^{2} \)
59 \( 1 + 1.08e6iT - 2.48e12T^{2} \)
61 \( 1 + 2.86e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.08e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.31e6iT - 9.09e12T^{2} \)
73 \( 1 - 1.75e5T + 1.10e13T^{2} \)
79 \( 1 - 1.62e6iT - 1.92e13T^{2} \)
83 \( 1 - 5.37e6T + 2.71e13T^{2} \)
89 \( 1 + 1.03e7T + 4.42e13T^{2} \)
97 \( 1 - 1.54e7iT - 8.07e13T^{2} \)
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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

−13.65592518208299414766543342867, −12.55577539578249243126622167787, −11.31420408602825204935125611310, −9.937197649811077777241855868724, −8.124775721222028196697122926566, −7.62271928100744529190637467751, −6.27440161532144202919553685015, −5.17886114823150782427218491764, −4.09529137472300728315289869224, −0.14153780711117330283266954439, 0.54323694024342480283623595941, 2.49854489024573858017593864203, 4.09273184051328081408960775509, 5.06221105057102491381802358479, 7.29680859759048667759458095099, 8.734667818681145583193862186598, 10.44807976104357250957296654598, 10.90208987109353172150140435256, 11.81537435005699704278251624179, 12.68016434332325459632421375138

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