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Properties

Label 2-84-12.11-c1-0-9
Degree $2$
Conductor $84$
Sign $0.372 + 0.927i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.349 − 1.37i)2-s + (1.72 + 0.203i)3-s + (−1.75 + 0.958i)4-s − 2.27i·5-s + (−0.323 − 2.42i)6-s + i·7-s + (1.92 + 2.06i)8-s + (2.91 + 0.699i)9-s + (−3.12 + 0.796i)10-s − 4.31·11-s + (−3.21 + 1.29i)12-s − 0.406·13-s + (1.37 − 0.349i)14-s + (0.463 − 3.91i)15-s + (2.16 − 3.36i)16-s + 4.31i·17-s + ⋯
L(s)  = 1  + (−0.247 − 0.968i)2-s + (0.993 + 0.117i)3-s + (−0.877 + 0.479i)4-s − 1.01i·5-s + (−0.131 − 0.991i)6-s + 0.377i·7-s + (0.681 + 0.731i)8-s + (0.972 + 0.233i)9-s + (−0.986 + 0.251i)10-s − 1.30·11-s + (−0.927 + 0.372i)12-s − 0.112·13-s + (0.366 − 0.0934i)14-s + (0.119 − 1.01i)15-s + (0.540 − 0.841i)16-s + 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.372 + 0.927i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.372 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849393 - 0.574020i\)
\(L(\frac12)\) \(\approx\) \(0.849393 - 0.574020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.349 + 1.37i)T \)
3 \( 1 + (-1.72 - 0.203i)T \)
7 \( 1 - iT \)
good5 \( 1 + 2.27iT - 5T^{2} \)
11 \( 1 + 4.31T + 11T^{2} \)
13 \( 1 + 0.406T + 13T^{2} \)
17 \( 1 - 4.31iT - 17T^{2} \)
19 \( 1 - 5.42iT - 19T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 + 3.72iT - 29T^{2} \)
31 \( 1 + 5.83iT - 31T^{2} \)
37 \( 1 + 7.83T + 37T^{2} \)
41 \( 1 - 2.56iT - 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + 2.32T + 47T^{2} \)
53 \( 1 + 5.48iT - 53T^{2} \)
59 \( 1 + 3.91T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + 7.83iT - 67T^{2} \)
71 \( 1 - 4.88T + 71T^{2} \)
73 \( 1 - 2.81T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 0.641T + 83T^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−13.72720244916694097779471348624, −12.88661283218103650883650722046, −12.22491786958803004923473089555, −10.54629571822235753086296957679, −9.675441879848941430984320756691, −8.488145298019278568451730508472, −7.960482561781990206096208100771, −5.22592067556480678949996007130, −3.80724863967067408017053414971, −2.08163777518713281410239250840, 3.00019171085947013424334186529, 4.90590518229268249723945115346, 6.85420002856986833498818634566, 7.45276404863617877655865107052, 8.673074881721534202761236489754, 9.888801881211033837564874993614, 10.80304478055909980747446199793, 12.88876994675212721382356739896, 13.81072111422622862672882914165, 14.47601160065199627150576747381

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