Properties

Label 2-168-7.2-c1-0-3
Degree $2$
Conductor $168$
Sign $0.928 + 0.371i$
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  + (0.5 + 0.866i)3-s + (2.13 − 3.70i)5-s + (−1.5 − 2.17i)7-s + (−0.499 + 0.866i)9-s + (2.13 + 3.70i)11-s − 1.27·13-s + 4.27·15-s + (2 + 3.46i)17-s + (−0.637 + 1.10i)19-s + (1.13 − 2.38i)21-s + (−2 + 3.46i)23-s + (−6.63 − 11.4i)25-s − 0.999·27-s − 2.27·29-s + (0.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.955 − 1.65i)5-s + (−0.566 − 0.823i)7-s + (−0.166 + 0.288i)9-s + (0.644 + 1.11i)11-s − 0.353·13-s + 1.10·15-s + (0.485 + 0.840i)17-s + (−0.146 + 0.253i)19-s + (0.248 − 0.521i)21-s + (−0.417 + 0.722i)23-s + (−1.32 − 2.29i)25-s − 0.192·27-s − 0.422·29-s + (0.0898 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.928 + 0.371i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.928 + 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32439 - 0.255412i\)
\(L(\frac12)\) \(\approx\) \(1.32439 - 0.255412i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.17i)T \)
good5 \( 1 + (-2.13 + 3.70i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.13 - 3.70i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.27T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.637 - 1.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.27T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.63 - 4.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 7.27T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.862 + 1.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.13 + 5.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.63 + 6.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (1.63 + 2.83i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.77 + 3.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.274T + 83T^{2} \)
89 \( 1 + (2.27 - 3.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−12.80981621443492262507128812764, −12.05038430443901154974108766302, −10.26604942530552231688799873964, −9.712858655875825753785361897452, −8.942470114228524780336869951277, −7.70210142937648354473962550165, −6.16650668961406270452541730571, −4.92512358477841411802329143516, −3.94681930248765243135654740900, −1.65272783622592330844999156291, 2.39625673710684810717946442515, 3.27195594437754236239031944103, 5.79316225419544392806601559543, 6.41263995154056942384939335563, 7.43415041361007257492982615890, 8.960436873339596611640659424711, 9.768976601911795217806400376679, 10.89164638736031302837718869850, 11.80990597597877025859883469302, 12.99381260311346479068755580646

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