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Properties

Label 2-168-168.125-c1-0-0
Degree $2$
Conductor $168$
Sign $0.242 - 0.970i$
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  − 1.41i·2-s + (−1.68 − 0.420i)3-s − 2.00·4-s + 3.91i·5-s + (−0.595 + 2.37i)6-s − 2.64·7-s + 2.82i·8-s + (2.64 + 1.41i)9-s + 5.53·10-s + (3.36 + 0.841i)12-s − 4.55·13-s + 3.74i·14-s + (1.64 − 6.57i)15-s + 4.00·16-s + (2 − 3.74i)18-s − 0.979·19-s + ⋯
L(s)  = 1  − 0.999i·2-s + (−0.970 − 0.242i)3-s − 1.00·4-s + 1.74i·5-s + (−0.242 + 0.970i)6-s − 0.999·7-s + 1.00i·8-s + (0.881 + 0.471i)9-s + 1.74·10-s + (0.970 + 0.242i)12-s − 1.26·13-s + 1.00i·14-s + (0.424 − 1.69i)15-s + 1.00·16-s + (0.471 − 0.881i)18-s − 0.224·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306943 + 0.239540i\)
\(L(\frac12)\) \(\approx\) \(0.306943 + 0.239540i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (1.68 + 0.420i)T \)
7 \( 1 + 2.64T \)
good5 \( 1 - 3.91iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.55T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 0.979T + 19T^{2} \)
23 \( 1 - 7.48iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 5.29iT - 59T^{2} \)
61 \( 1 - 15.6T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 5.29T + 79T^{2} \)
83 \( 1 - 18.1iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.76990411919402532981323281943, −11.80681540511892720265038466820, −11.09332722259486800288715215324, −10.13847854126621573192967799931, −9.674071931235923495015654491147, −7.55648679424282095323554939957, −6.66547016644415000461889342567, −5.45358464703962491850179723483, −3.72701974364572639745311570147, −2.45235352092706127272020359547, 0.40116088426021945553881671768, 4.25294849207653958435038262608, 5.03857142588052947880103222464, 6.02988101934568946142342878964, 7.15234891862770925746923922186, 8.539737187818143120668098223062, 9.463963209256734084882883781933, 10.19226691641933905765379508982, 12.09530534724240919172864824550, 12.62666032474629102381188587931

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