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Properties

Label 2-168-168.101-c1-0-10
Degree $2$
Conductor $168$
Sign $0.706 - 0.707i$
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 + (3.62 − 2.09i)5-s − 2.44i·6-s + (1.62 + 2.09i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (5.12 + 2.95i)10-s + (−0.0857 + 0.148i)11-s + (2.99 − 1.73i)12-s + (−1.41 + 3.46i)14-s − 7.24·15-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 + (1.61 − 0.935i)5-s − 0.999i·6-s + (0.612 + 0.790i)7-s − 0.999·8-s + (0.5 + 0.866i)9-s + (1.61 + 0.935i)10-s + (−0.0258 + 0.0448i)11-s + (0.866 − 0.500i)12-s + (−0.377 + 0.925i)14-s − 1.87·15-s + (−0.500 − 0.866i)16-s + (−0.499 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.706 - 0.707i$
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: \((2,\ 168,\ (\ :1/2),\ 0.706 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27016 + 0.526828i\)
\(L(\frac12)\) \(\approx\) \(1.27016 + 0.526828i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
good5 \( 1 + (-3.62 + 2.09i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.0857 - 0.148i)T + (-5.5 - 9.52i)T^{2} \)
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 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + (5.37 + 3.10i)T + (15.5 + 26.8i)T^{2} \)
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 \( 1 + (5.03 - 8.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.98 + 2.30i)T + (29.5 + 51.0i)T^{2} \)
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 \( 1 + (3.86 + 6.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.76iT - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.5iT - 97T^{2} \)
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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

−12.93456247042345658165356284819, −12.38535865269789043893901814777, −11.19218981657080973868170183478, −9.625073846918497790010139678895, −8.771094644198578161487309569184, −7.52877949431082005621226028509, −6.08847587932599529403997670380, −5.58776190811996915368453182123, −4.73995491413624673763265631232, −1.97590010121328969886745868285, 1.77186158140741299978715041944, 3.57254652042876278285779701688, 5.06325435010082705782695807136, 5.89745629144618398584096662281, 6.98257942465056893820267452275, 9.286755719502426704515534792916, 10.01596897222987889235067822452, 10.83523014377108732438766242428, 11.25776208877570194586715494019, 12.72718229881960371213861266279

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