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Properties

Label 2-168-168.101-c1-0-0
Degree $2$
Conductor $168$
Sign $0.00925 - 0.999i$
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  + (−0.707 − 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (−0.621 + 0.358i)5-s + 2.44i·6-s + (−2.62 − 0.358i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (0.878 + 0.507i)10-s + (−2.91 + 5.04i)11-s + (2.99 − 1.73i)12-s + (1.41 + 3.46i)14-s + 1.24·15-s + (−2.00 − 3.46i)16-s + (2.12 − 3.67i)18-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−0.866 − 0.499i)3-s + (−0.499 + 0.866i)4-s + (−0.277 + 0.160i)5-s + 0.999i·6-s + (−0.990 − 0.135i)7-s + 0.999·8-s + (0.5 + 0.866i)9-s + (0.277 + 0.160i)10-s + (−0.878 + 1.52i)11-s + (0.866 − 0.500i)12-s + (0.377 + 0.925i)14-s + 0.320·15-s + (−0.500 − 0.866i)16-s + (0.499 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00925 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00925 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.00925 - 0.999i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.00925 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113618 + 0.112572i\)
\(L(\frac12)\) \(\approx\) \(0.113618 + 0.112572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (2.62 + 0.358i)T \)
good5 \( 1 + (0.621 - 0.358i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.91 - 5.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.58T + 29T^{2} \)
31 \( 1 + (9.62 + 5.55i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.03 + 3.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-12.9 - 7.49i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (8.48 + 4.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.86 - 15.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.5iT - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.06iT - 97T^{2} \)
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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

−12.97784463055286492133030372230, −12.01448529434204898744187663661, −11.06623529525368955250454568354, −10.16835078161391702408581425150, −9.377064313900649788573689064210, −7.63798360644569760032079635865, −7.11434216307483274562168947597, −5.41072417612169412885880584733, −3.93336120093502218669534907081, −2.14401427075237738812491018014, 0.18704754528549063127061355071, 3.70829686570243310677140819631, 5.34860888934293696943476691805, 6.01063201128961134553312901332, 7.17665805796150911591748976546, 8.511580794288346771858191673026, 9.463437604897872984536497197881, 10.47340527578080963652480133963, 11.23145690148664634458930515722, 12.64339330641325507128611427793

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