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Properties

Label 2-84-7.2-c3-0-1
Degree $2$
Conductor $84$
Sign $0.502 - 0.864i$
Analytic cond. $4.95616$
Root an. cond. $2.22624$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 2.59i)3-s + (−6.22 + 10.7i)5-s + (15.3 + 10.2i)7-s + (−4.5 + 7.79i)9-s + (25.5 + 44.2i)11-s + 37.2·13-s + 37.3·15-s + (−11.1 − 19.2i)17-s + (−27.1 + 47.0i)19-s + (3.66 − 55.4i)21-s + (−88.4 + 153. i)23-s + (−14.9 − 25.9i)25-s + 27·27-s + 61.0·29-s + (−159. − 277. i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.556 + 0.964i)5-s + (0.831 + 0.556i)7-s + (−0.166 + 0.288i)9-s + (0.700 + 1.21i)11-s + 0.794·13-s + 0.642·15-s + (−0.158 − 0.274i)17-s + (−0.328 + 0.568i)19-s + (0.0381 − 0.576i)21-s + (−0.801 + 1.38i)23-s + (−0.119 − 0.207i)25-s + 0.192·27-s + 0.391·29-s + (−0.926 − 1.60i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.502 - 0.864i$
Analytic conductor: \(4.95616\)
Root analytic conductor: \(2.22624\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :3/2),\ 0.502 - 0.864i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.11481 + 0.641620i\)
\(L(\frac12)\) \(\approx\) \(1.11481 + 0.641620i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 + (-15.3 - 10.2i)T \)
good5 \( 1 + (6.22 - 10.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-25.5 - 44.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 37.2T + 2.19e3T^{2} \)
17 \( 1 + (11.1 + 19.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (27.1 - 47.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (88.4 - 153. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 61.0T + 2.43e4T^{2} \)
31 \( 1 + (159. + 277. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-157. + 272. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 206.T + 6.89e4T^{2} \)
43 \( 1 - 339.T + 7.95e4T^{2} \)
47 \( 1 + (71.0 - 123. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (155. + 268. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-140. - 243. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-271. + 470. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-239. - 415. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.10e3T + 3.57e5T^{2} \)
73 \( 1 + (119. + 207. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (580. - 1.00e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 2.93T + 5.71e5T^{2} \)
89 \( 1 + (-639. + 1.10e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 79.0T + 9.12e5T^{2} \)
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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

−14.15198762875072509288277694282, −12.71506346191371331179828193081, −11.62920184662554254762612314609, −11.08138197139312865830170306432, −9.544096545418988071152647667057, −8.007761348590916812435360310504, −7.10204268317453016836712492321, −5.78835256896650490276250342844, −3.99985911549111170670009956791, −1.95768556703895294904276176935, 0.895062648107257880001035876520, 3.85246432158420167211907892682, 4.87648557118163796343368325133, 6.41534323992155094725804974663, 8.285602343179141403045804233851, 8.808011714753263927219064747200, 10.56300826716942966524317389842, 11.34439871177909027122641640232, 12.37093906461263663191788528935, 13.68320415243728528256396048780

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