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Properties

Label 2-84-12.11-c5-0-12
Degree $2$
Conductor $84$
Sign $-0.781 - 0.624i$
Analytic cond. $13.4722$
Root an. cond. $3.67045$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.35 + 1.82i)2-s + (−15.1 + 3.68i)3-s + (25.3 − 19.5i)4-s + 66.1i·5-s + (74.3 − 47.4i)6-s − 49i·7-s + (−99.7 + 151. i)8-s + (215. − 111. i)9-s + (−120. − 354. i)10-s + 653.·11-s + (−311. + 389. i)12-s + 260.·13-s + (89.5 + 262. i)14-s + (−243. − 1.00e3i)15-s + (257. − 991. i)16-s − 5.25i·17-s + ⋯
L(s)  = 1  + (−0.946 + 0.323i)2-s + (−0.971 + 0.236i)3-s + (0.791 − 0.611i)4-s + 1.18i·5-s + (0.843 − 0.537i)6-s − 0.377i·7-s + (−0.551 + 0.834i)8-s + (0.888 − 0.459i)9-s + (−0.382 − 1.11i)10-s + 1.62·11-s + (−0.624 + 0.781i)12-s + 0.427·13-s + (0.122 + 0.357i)14-s + (−0.279 − 1.14i)15-s + (0.251 − 0.967i)16-s − 0.00441i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.781 - 0.624i$
Analytic conductor: \(13.4722\)
Root analytic conductor: \(3.67045\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :5/2),\ -0.781 - 0.624i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.223199 + 0.637079i\)
\(L(\frac12)\) \(\approx\) \(0.223199 + 0.637079i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.35 - 1.82i)T \)
3 \( 1 + (15.1 - 3.68i)T \)
7 \( 1 + 49iT \)
good5 \( 1 - 66.1iT - 3.12e3T^{2} \)
11 \( 1 - 653.T + 1.61e5T^{2} \)
13 \( 1 - 260.T + 3.71e5T^{2} \)
17 \( 1 + 5.25iT - 1.41e6T^{2} \)
19 \( 1 - 2.67e3iT - 2.47e6T^{2} \)
23 \( 1 + 4.64e3T + 6.43e6T^{2} \)
29 \( 1 + 1.16e3iT - 2.05e7T^{2} \)
31 \( 1 + 281. iT - 2.86e7T^{2} \)
37 \( 1 + 1.87e3T + 6.93e7T^{2} \)
41 \( 1 - 1.25e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.81e4iT - 1.47e8T^{2} \)
47 \( 1 - 9.83e3T + 2.29e8T^{2} \)
53 \( 1 + 2.71e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.03e4T + 7.14e8T^{2} \)
61 \( 1 + 8.68e3T + 8.44e8T^{2} \)
67 \( 1 + 6.10e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.74e4T + 1.80e9T^{2} \)
73 \( 1 + 6.84e4T + 2.07e9T^{2} \)
79 \( 1 - 7.56e4iT - 3.07e9T^{2} \)
83 \( 1 - 5.85e4T + 3.93e9T^{2} \)
89 \( 1 - 3.02e4iT - 5.58e9T^{2} \)
97 \( 1 + 7.48e4T + 8.58e9T^{2} \)
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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

−14.18170156957362024384319735577, −12.07696275826593872474542407312, −11.32539046700915773402133183482, −10.34692321339590409430170336364, −9.621783976373133757667499667566, −7.88735840792377773766398475859, −6.58117344759527883063769874338, −6.08274393259955855526905428455, −3.83529925293942552130139127711, −1.43631814534852605663930157851, 0.47922271355392630691099685477, 1.66483469869168392189802234248, 4.23568271351250942067964139512, 5.91157050429856217844286070332, 7.08850811491920105262065544511, 8.648285314143350976447873288491, 9.367101803730957443362329687043, 10.76955176011419992309372930082, 11.94626072260756352508103150891, 12.25182473042320115552260739797

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