Properties

Label 2-168-168.125-c1-0-10
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

\(L(1)\) \(\approx\) \(0.812516 + 1.04114i\)
\(L(\frac12)\) \(\approx\) \(0.812516 + 1.04114i\)
\(L(\frac{3}{2})\) 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

−13.55082429441415591409002296235, −12.50221530797643592391357001073, −10.74265430773582338770900949019, −9.895123011417257168027292492298, −8.840629193448992531532894692301, −7.72884113914535507680162903545, −6.70838748321316437944352951414, −6.24246170687468710128333134418, −3.89135341580625531500095494698, −2.94718055445782401687889657671, 1.42638230889915777891905303173, 3.34275970871489480947168112588, 4.31870903147471137412630187115, 5.63570856612103873034045206863, 7.84834102385740257855322544008, 8.912533024591333633698434162696, 9.266482582456528696222940653048, 10.26744980315384565317696139653, 11.69523619649111839271029696122, 12.74811382195410214396024616967

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