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Properties

Label 2-84-28.3-c1-0-5
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: $\chi_{84} (31, \cdot )$
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

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

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