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Properties

Label 2-84-12.11-c1-0-0
Degree $2$
Conductor $84$
Sign $0.655 - 0.755i$
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  + (−0.750 − 1.19i)2-s + (−0.448 + 1.67i)3-s + (−0.874 + 1.79i)4-s + 3.56i·5-s + (2.34 − 0.717i)6-s − i·7-s + (2.81 − 0.301i)8-s + (−2.59 − 1.50i)9-s + (4.27 − 2.67i)10-s + 0.335·11-s + (−2.61 − 2.26i)12-s + 3.34·13-s + (−1.19 + 0.750i)14-s + (−5.96 − 1.59i)15-s + (−2.47 − 3.14i)16-s + 0.335i·17-s + ⋯
L(s)  = 1  + (−0.530 − 0.847i)2-s + (−0.258 + 0.965i)3-s + (−0.437 + 0.899i)4-s + 1.59i·5-s + (0.956 − 0.292i)6-s − 0.377i·7-s + (0.994 − 0.106i)8-s + (−0.865 − 0.500i)9-s + (1.35 − 0.845i)10-s + 0.101·11-s + (−0.755 − 0.655i)12-s + 0.928·13-s + (−0.320 + 0.200i)14-s + (−1.53 − 0.412i)15-s + (−0.617 − 0.786i)16-s + 0.0813i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.655 - 0.755i$
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: \((2,\ 84,\ (\ :1/2),\ 0.655 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.609262 + 0.278152i\)
\(L(\frac12)\) \(\approx\) \(0.609262 + 0.278152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.750 + 1.19i)T \)
3 \( 1 + (0.448 - 1.67i)T \)
7 \( 1 + iT \)
good5 \( 1 - 3.56iT - 5T^{2} \)
11 \( 1 - 0.335T + 11T^{2} \)
13 \( 1 - 3.34T + 13T^{2} \)
17 \( 1 - 0.335iT - 17T^{2} \)
19 \( 1 - 1.84iT - 19T^{2} \)
23 \( 1 - 4.45T + 23T^{2} \)
29 \( 1 + 5.91iT - 29T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 - 1.45iT - 41T^{2} \)
43 \( 1 - 7.49iT - 43T^{2} \)
47 \( 1 - 8.91T + 47T^{2} \)
53 \( 1 + 4.79iT - 53T^{2} \)
59 \( 1 + 14.0T + 59T^{2} \)
61 \( 1 + 0.353T + 61T^{2} \)
67 \( 1 + 3.19iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 4.69T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 6.89T + 83T^{2} \)
89 \( 1 + 3.87iT - 89T^{2} \)
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

−14.44560357915784377474319542498, −13.38435362051140736169804635234, −11.67400715446218811188935528182, −10.95985598067422645482632690547, −10.31578174825327489979037355894, −9.288367238494858647761631663946, −7.78276230311544649231944893930, −6.26621205603552343560941257059, −4.10712844353145207623894948623, −2.98550399227164041205278559022, 1.22071947549038401216141071232, 4.93038863201013648224285237350, 5.92686409773434031734966823308, 7.29161054844370346481905777252, 8.629199358404285528795925861595, 8.986163172981130285761828831775, 10.87569475910477147452375774802, 12.25076872741603559971806216722, 13.13403303387676268883641638676, 13.98551726114589532365316399753

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