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Properties

Label 2-168-168.5-c1-0-7
Degree $2$
Conductor $168$
Sign $0.999 - 0.00700i$
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  + (−0.174 − 1.40i)2-s + (1.09 + 1.34i)3-s + (−1.93 + 0.489i)4-s + (2.46 + 1.42i)5-s + (1.69 − 1.76i)6-s + (−1.02 + 2.44i)7-s + (1.02 + 2.63i)8-s + (−0.614 + 2.93i)9-s + (1.57 − 3.71i)10-s + (−2.42 − 4.20i)11-s + (−2.77 − 2.07i)12-s + 2.75·13-s + (3.60 + 1.00i)14-s + (0.780 + 4.87i)15-s + (3.52 − 1.89i)16-s + (−1.75 − 3.03i)17-s + ⋯
L(s)  = 1  + (−0.123 − 0.992i)2-s + (0.630 + 0.776i)3-s + (−0.969 + 0.244i)4-s + (1.10 + 0.637i)5-s + (0.692 − 0.721i)6-s + (−0.385 + 0.922i)7-s + (0.362 + 0.931i)8-s + (−0.204 + 0.978i)9-s + (0.496 − 1.17i)10-s + (−0.731 − 1.26i)11-s + (−0.801 − 0.598i)12-s + 0.763·13-s + (0.963 + 0.268i)14-s + (0.201 + 1.25i)15-s + (0.880 − 0.474i)16-s + (−0.425 − 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.999 - 0.00700i$
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: \((2,\ 168,\ (\ :1/2),\ 0.999 - 0.00700i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30798 + 0.00458005i\)
\(L(\frac12)\) \(\approx\) \(1.30798 + 0.00458005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.174 + 1.40i)T \)
3 \( 1 + (-1.09 - 1.34i)T \)
7 \( 1 + (1.02 - 2.44i)T \)
good5 \( 1 + (-2.46 - 1.42i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.42 + 4.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.75T + 13T^{2} \)
17 \( 1 + (1.75 + 3.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.14 + 5.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.15 + 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.90T + 29T^{2} \)
31 \( 1 + (0.858 - 0.495i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.06 - 0.614i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.10T + 41T^{2} \)
43 \( 1 - 5.11iT - 43T^{2} \)
47 \( 1 + (5.61 - 9.72i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.00 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.890 - 0.514i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.24 + 2.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 2.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.75iT - 71T^{2} \)
73 \( 1 + (-0.291 + 0.168i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.80 - 4.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.138iT - 83T^{2} \)
89 \( 1 + (0.580 - 1.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−13.03464341573997116709456292646, −11.41616560348109131263861017435, −10.77103196358853908858373859188, −9.751854858263090076152424851972, −9.105277010721157580297198799791, −8.191503434074307025402371796279, −6.08865146935374269864767448676, −4.98607615844171756632715325076, −3.13829409214304153619179623152, −2.52132755699143949528029524624, 1.57900902465625109773334741940, 3.95284831067853628560032509104, 5.54480241260678838011553556026, 6.55911158969156425161108099610, 7.58108965671230040690927359613, 8.482526064608429783630863263506, 9.658529845256531023518966744484, 10.19983566971043299989533481369, 12.42263731563898376072620063718, 13.17370495002451205857101203119

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