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Properties

Label 2-168-168.5-c1-0-24
Degree $2$
Conductor $168$
Sign $0.00925 + 0.999i$
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  + (0.707 − 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (0.621 + 0.358i)5-s − 2.44i·6-s + (−2.62 + 0.358i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (0.878 − 0.507i)10-s + (2.91 + 5.04i)11-s + (−2.99 − 1.73i)12-s + (−1.41 + 3.46i)14-s + 1.24·15-s + (−2.00 + 3.46i)16-s + (−2.12 − 3.67i)18-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)2-s + (0.866 − 0.499i)3-s + (−0.499 − 0.866i)4-s + (0.277 + 0.160i)5-s − 0.999i·6-s + (−0.990 + 0.135i)7-s − 0.999·8-s + (0.5 − 0.866i)9-s + (0.277 − 0.160i)10-s + (0.878 + 1.52i)11-s + (−0.866 − 0.500i)12-s + (−0.377 + 0.925i)14-s + 0.320·15-s + (−0.500 + 0.866i)16-s + (−0.499 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00925 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00925 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.00925 + 0.999i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.00925 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21081 - 1.19966i\)
\(L(\frac12)\) \(\approx\) \(1.21081 - 1.19966i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (2.62 - 0.358i)T \)
good5 \( 1 + (-0.621 - 0.358i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.91 - 5.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.58T + 29T^{2} \)
31 \( 1 + (9.62 - 5.55i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.03 + 3.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.9 - 7.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (8.48 - 4.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.86 + 15.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.5iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.06iT - 97T^{2} \)
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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

−12.46618357453874285774754724401, −12.07795045395855681803578194940, −10.36027783826117806662156251383, −9.607895928041710259448305815718, −8.862718452280285750200296245631, −7.12005730994234531843561402131, −6.20966689603491368244029360232, −4.39130656411964321773884536855, −3.15682033607213618155360099238, −1.85612458809570943607386661198, 3.12843816523757430837690512104, 4.00069423234993990684982530594, 5.58607104680017025375238457088, 6.62971602430470037355028976819, 7.916220613973956293942346581565, 8.984240680934966648402970104489, 9.535930006267896807775903128464, 11.03318692728727745205172008456, 12.46008047148235720539754253377, 13.49644490305673902599548191684

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