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Properties

Label 2-84-28.19-c1-0-3
Degree $2$
Conductor $84$
Sign $0.945 - 0.324i$
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  + (1.26 + 0.638i)2-s + (−0.5 − 0.866i)3-s + (1.18 + 1.61i)4-s + (0.380 + 0.219i)5-s + (−0.0777 − 1.41i)6-s + (−2.02 − 1.70i)7-s + (0.464 + 2.79i)8-s + (−0.499 + 0.866i)9-s + (0.339 + 0.519i)10-s + (−1.83 + 1.05i)11-s + (0.803 − 1.83i)12-s − 3.84i·13-s + (−1.46 − 3.44i)14-s − 0.438i·15-s + (−1.19 + 3.81i)16-s + (−4.89 + 2.82i)17-s + ⋯
L(s)  = 1  + (0.892 + 0.451i)2-s + (−0.288 − 0.499i)3-s + (0.592 + 0.805i)4-s + (0.170 + 0.0981i)5-s + (−0.0317 − 0.576i)6-s + (−0.764 − 0.644i)7-s + (0.164 + 0.986i)8-s + (−0.166 + 0.288i)9-s + (0.107 + 0.164i)10-s + (−0.552 + 0.318i)11-s + (0.232 − 0.528i)12-s − 1.06i·13-s + (−0.391 − 0.920i)14-s − 0.113i·15-s + (−0.299 + 0.954i)16-s + (−1.18 + 0.684i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.945 - 0.324i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.945 - 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30937 + 0.218274i\)
\(L(\frac12)\) \(\approx\) \(1.30937 + 0.218274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.638i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.02 + 1.70i)T \)
good5 \( 1 + (-0.380 - 0.219i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.83 - 1.05i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.84iT - 13T^{2} \)
17 \( 1 + (4.89 - 2.82i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.48 + 2.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.13 - 2.38i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.02T + 29T^{2} \)
31 \( 1 + (-3.71 - 6.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.64 + 4.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.81iT - 41T^{2} \)
43 \( 1 - 4.38iT - 43T^{2} \)
47 \( 1 + (-0.844 + 1.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.35 + 9.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.05 + 7.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.35 - 3.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.79 + 3.92i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.16iT - 71T^{2} \)
73 \( 1 + (8.69 - 5.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.4 + 7.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.49T + 83T^{2} \)
89 \( 1 + (9.02 + 5.20i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.22iT - 97T^{2} \)
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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

−14.12419542977144116676208420424, −13.10839647346898758757573051491, −12.73412384741715867269266847913, −11.27671800140568837115877842674, −10.21053657525708806360020196963, −8.295618089659163629589658388513, −7.08787257613416308919925130585, −6.19148648268863669786793614894, −4.74195767023754797170860535265, −2.93046127186041610174628907038, 2.73646517998044015543004009300, 4.40558674575738670170015195993, 5.69223575062317852967504118121, 6.78587590863770028962101775216, 9.021160474299183535693520809658, 9.983048142354471465974763990626, 11.20264449354919681737415742884, 12.04289346586973675320783443199, 13.17979718492096294311557233550, 14.02653526872965812575454138374

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