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Properties

Label 2-168-56.3-c1-0-3
Degree $2$
Conductor $168$
Sign $-0.410 - 0.911i$
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  + (0.221 + 1.39i)2-s + (0.866 + 0.5i)3-s + (−1.90 + 0.617i)4-s + (0.225 + 0.390i)5-s + (−0.507 + 1.32i)6-s + (−0.458 + 2.60i)7-s + (−1.28 − 2.52i)8-s + (0.499 + 0.866i)9-s + (−0.495 + 0.401i)10-s + (−0.360 + 0.623i)11-s + (−1.95 − 0.416i)12-s + 3.48·13-s + (−3.74 − 0.0641i)14-s + 0.451i·15-s + (3.23 − 2.34i)16-s + (−3.55 − 2.05i)17-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (0.499 + 0.288i)3-s + (−0.951 + 0.308i)4-s + (0.100 + 0.174i)5-s + (−0.206 + 0.538i)6-s + (−0.173 + 0.984i)7-s + (−0.453 − 0.891i)8-s + (0.166 + 0.288i)9-s + (−0.156 + 0.126i)10-s + (−0.108 + 0.188i)11-s + (−0.564 − 0.120i)12-s + 0.967·13-s + (−0.999 − 0.0171i)14-s + 0.116i·15-s + (0.809 − 0.587i)16-s + (−0.862 − 0.498i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.410 - 0.911i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.410 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704645 + 1.09051i\)
\(L(\frac12)\) \(\approx\) \(0.704645 + 1.09051i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.221 - 1.39i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.458 - 2.60i)T \)
good5 \( 1 + (-0.225 - 0.390i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.360 - 0.623i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + (3.55 + 2.05i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.97 + 2.29i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0459 + 0.0265i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.85iT - 29T^{2} \)
31 \( 1 + (4.58 - 7.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.51 + 4.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.94iT - 41T^{2} \)
43 \( 1 - 5.17T + 43T^{2} \)
47 \( 1 + (-0.460 - 0.796i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.71 + 1.56i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.86 + 2.80i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.54 + 4.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.93 - 8.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + (3.33 + 1.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.40 - 4.85i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.53iT - 83T^{2} \)
89 \( 1 + (12.6 - 7.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.14iT - 97T^{2} \)
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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

−13.38357347386695341698463438549, −12.41995852982402473601607609293, −11.11276623742852438700628833936, −9.624233735661235413681540005718, −8.962943174106604424030507399210, −8.017834252071636353668152527614, −6.75100797102302425821448601714, −5.68275373984258330708553354950, −4.43517030377753956623354581897, −2.89671662780268044300387028796, 1.37841628136049024671613668682, 3.23001532832955365449882033399, 4.27230703134949571423018271860, 5.88195723680206667020632076573, 7.42709391550978124860431380797, 8.631989708638726250827305196535, 9.514233271069988349886644807883, 10.65054519558561786642371957017, 11.34951296015502449061295448164, 12.74091162790917764175059853256

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