Properties

Label 2-168-168.5-c1-0-12
Degree $2$
Conductor $168$
Sign $0.0957 - 0.995i$
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  + (0.0857 + 1.41i)2-s + (1.69 + 0.366i)3-s + (−1.98 + 0.242i)4-s + (2.66 + 1.54i)5-s + (−0.372 + 2.42i)6-s + (−1.46 − 2.20i)7-s + (−0.511 − 2.78i)8-s + (2.73 + 1.24i)9-s + (−1.94 + 3.89i)10-s + (−0.621 − 1.07i)11-s + (−3.44 − 0.318i)12-s − 5.98·13-s + (2.98 − 2.25i)14-s + (3.95 + 3.58i)15-s + (3.88 − 0.960i)16-s + (0.595 + 1.03i)17-s + ⋯
L(s)  = 1  + (0.0606 + 0.998i)2-s + (0.977 + 0.211i)3-s + (−0.992 + 0.121i)4-s + (1.19 + 0.688i)5-s + (−0.152 + 0.988i)6-s + (−0.552 − 0.833i)7-s + (−0.180 − 0.983i)8-s + (0.910 + 0.414i)9-s + (−0.615 + 1.23i)10-s + (−0.187 − 0.324i)11-s + (−0.995 − 0.0920i)12-s − 1.66·13-s + (0.798 − 0.602i)14-s + (1.02 + 0.926i)15-s + (0.970 − 0.240i)16-s + (0.144 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0957 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0957 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.0957 - 0.995i$
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: \((2,\ 168,\ (\ :1/2),\ 0.0957 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13790 + 1.03366i\)
\(L(\frac12)\) \(\approx\) \(1.13790 + 1.03366i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0857 - 1.41i)T \)
3 \( 1 + (-1.69 - 0.366i)T \)
7 \( 1 + (1.46 + 2.20i)T \)
good5 \( 1 + (-2.66 - 1.54i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.621 + 1.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.98T + 13T^{2} \)
17 \( 1 + (-0.595 - 1.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.614 + 1.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.56 + 1.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.19T + 29T^{2} \)
31 \( 1 + (-1.33 + 0.773i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.334 + 0.193i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.44T + 41T^{2} \)
43 \( 1 + 8.29iT - 43T^{2} \)
47 \( 1 + (-3.34 + 5.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.25 - 9.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.22 - 1.86i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.16 + 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.7 - 6.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.21iT - 71T^{2} \)
73 \( 1 + (8.92 - 5.15i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.41 - 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.22iT - 83T^{2} \)
89 \( 1 + (6.94 - 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−13.48121572414361865245712883692, −12.59904419237282548141913315532, −10.34839354503403884414710988285, −9.919212152265186950404794782491, −9.016769800950299651740556021534, −7.59121981069069050082670022589, −6.93996238044928832168874163511, −5.65703079679624852336279359824, −4.17430984920452054806581695253, −2.67825410238353480102841831450, 1.92604965758628772659653896428, 2.85720545392912396130966329342, 4.66693758864834600289615720872, 5.83199516111567949942628110754, 7.66147835900374497053093603161, 8.993503056162948148264820729290, 9.564077373432417894210749854087, 10.09465571923622014919948980214, 11.97008737398970704778166157662, 12.67430146798281752823836461251

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