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Properties

Label 2-84-7.2-c1-0-0
Degree $2$
Conductor $84$
Sign $0.968 - 0.250i$
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  + (0.5 + 0.866i)3-s + (1 − 1.73i)5-s + (0.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s − 3·13-s + 1.99·15-s + (−4 − 6.92i)17-s + (0.5 − 0.866i)19-s + (−2 + 1.73i)21-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s − 0.999·27-s + 4·29-s + (−1.5 − 2.59i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.447 − 0.774i)5-s + (0.188 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s − 0.832·13-s + 0.516·15-s + (−0.970 − 1.68i)17-s + (0.114 − 0.198i)19-s + (−0.436 + 0.377i)21-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s − 0.192·27-s + 0.742·29-s + (−0.269 − 0.466i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05425 + 0.134347i\)
\(L(\frac12)\) \(\approx\) \(1.05425 + 0.134347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (4 + 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−14.25852321186986371167230880401, −13.36824278900699912279613894739, −12.16412271182825664677552101971, −11.14486027425762165114939820339, −9.491877347068163021800344931572, −9.081300569046940718317926347947, −7.66253600109065687526637660338, −5.71540711166325100898174515524, −4.74540548980467941915097919083, −2.59165756938551738616972537442, 2.31090490712168537397656940994, 4.27368775553655232306587359177, 6.30079322587263930132470688455, 7.24475308339824837646988399442, 8.435133764296601594801084367096, 10.12769028832827674560627736585, 10.69561359373933840087034095162, 12.29757996768351210651907455842, 13.23173806785161329894210906434, 14.34877846489454647415739956355

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