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Properties

Label 2-168-21.5-c1-0-7
Degree $2$
Conductor $168$
Sign $0.446 + 0.894i$
Analytic cond. $1.34148$
Root an. cond. $1.15822$
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  + (0.642 − 1.60i)3-s + (1.28 − 2.23i)5-s + (−0.203 + 2.63i)7-s + (−2.17 − 2.06i)9-s + (−1.43 + 0.826i)11-s − 5.71i·13-s + (−2.76 − 3.50i)15-s + (3.79 + 6.56i)17-s + (2.58 + 1.49i)19-s + (4.11 + 2.02i)21-s + (−0.249 − 0.143i)23-s + (−0.825 − 1.43i)25-s + (−4.72 + 2.16i)27-s + 2.05i·29-s + (−5.21 + 3.00i)31-s + ⋯
L(s)  = 1  + (0.371 − 0.928i)3-s + (0.576 − 0.998i)5-s + (−0.0768 + 0.997i)7-s + (−0.724 − 0.689i)9-s + (−0.431 + 0.249i)11-s − 1.58i·13-s + (−0.713 − 0.906i)15-s + (0.919 + 1.59i)17-s + (0.594 + 0.343i)19-s + (0.897 + 0.441i)21-s + (−0.0519 − 0.0300i)23-s + (−0.165 − 0.286i)25-s + (−0.908 + 0.417i)27-s + 0.382i·29-s + (−0.936 + 0.540i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.446 + 0.894i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.446 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12886 - 0.697902i\)
\(L(\frac12)\) \(\approx\) \(1.12886 - 0.697902i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.642 + 1.60i)T \)
7 \( 1 + (0.203 - 2.63i)T \)
good5 \( 1 + (-1.28 + 2.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.43 - 0.826i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.71iT - 13T^{2} \)
17 \( 1 + (-3.79 - 6.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.58 - 1.49i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.249 + 0.143i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.05iT - 29T^{2} \)
31 \( 1 + (5.21 - 3.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.877 - 1.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.28T + 41T^{2} \)
43 \( 1 - 2.46T + 43T^{2} \)
47 \( 1 + (0.186 - 0.323i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.73 - 3.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.89 - 8.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.889 - 0.513i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.18 + 2.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (3.30 - 1.90i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.56 + 7.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.65T + 83T^{2} \)
89 \( 1 + (-7.25 + 12.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.43iT - 97T^{2} \)
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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

−12.65014217172046257877717756738, −12.19725621120459483666476153929, −10.54633673476420683145203227944, −9.350370611220615165557406338531, −8.428971023304529311347938692501, −7.69158529531636780839245793654, −5.93024605150563191657930525328, −5.40094976398656786488594653378, −3.12830969473408613093517874603, −1.54599971558872930799597257565, 2.64299377343995631968414634571, 3.90246754282334564906413423822, 5.24882544136215794032989552442, 6.74683927650774192148385435272, 7.69682599209965998960956957480, 9.346904581938610826027523096297, 9.851839630098720146248080482926, 10.89778576100857752246251646829, 11.56357135658222595815385031123, 13.40465817184671209409115843717

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