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Properties

Label 2-168-168.107-c1-0-23
Degree $2$
Conductor $168$
Sign $0.627 + 0.778i$
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  + (0.982 − 1.01i)2-s + (1.72 − 0.143i)3-s + (−0.0680 − 1.99i)4-s + (−1.77 + 3.07i)5-s + (1.55 − 1.89i)6-s + (0.793 − 2.52i)7-s + (−2.09 − 1.89i)8-s + (2.95 − 0.496i)9-s + (1.38 + 4.83i)10-s + (0.396 − 0.229i)11-s + (−0.404 − 3.44i)12-s + 0.799i·13-s + (−1.78 − 3.28i)14-s + (−2.62 + 5.57i)15-s + (−3.99 + 0.271i)16-s + (−5.48 + 3.16i)17-s + ⋯
L(s)  = 1  + (0.694 − 0.719i)2-s + (0.996 − 0.0829i)3-s + (−0.0340 − 0.999i)4-s + (−0.795 + 1.37i)5-s + (0.632 − 0.774i)6-s + (0.299 − 0.953i)7-s + (−0.742 − 0.670i)8-s + (0.986 − 0.165i)9-s + (0.437 + 1.52i)10-s + (0.119 − 0.0690i)11-s + (−0.116 − 0.993i)12-s + 0.221i·13-s + (−0.477 − 0.878i)14-s + (−0.678 + 1.43i)15-s + (−0.997 + 0.0679i)16-s + (−1.33 + 0.767i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.627 + 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69461 - 0.810752i\)
\(L(\frac12)\) \(\approx\) \(1.69461 - 0.810752i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.982 + 1.01i)T \)
3 \( 1 + (-1.72 + 0.143i)T \)
7 \( 1 + (-0.793 + 2.52i)T \)
good5 \( 1 + (1.77 - 3.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.396 + 0.229i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.799iT - 13T^{2} \)
17 \( 1 + (5.48 - 3.16i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.61 - 4.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.55 - 2.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.60T + 29T^{2} \)
31 \( 1 + (-4.42 + 2.55i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.89 + 1.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 - 3.56T + 43T^{2} \)
47 \( 1 + (-2.15 + 3.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.16 + 2.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.49 + 0.864i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.60 - 2.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.00979 - 0.0169i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.04T + 71T^{2} \)
73 \( 1 + (-4.15 - 7.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.8 + 6.28i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.694iT - 83T^{2} \)
89 \( 1 + (7.02 + 4.05i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.05T + 97T^{2} \)
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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

−12.76957545583174284833364599013, −11.58590241108170060936229866850, −10.67266034949331695632724170503, −10.08984775639729746899907077199, −8.533022001802395382684672653254, −7.31876917980057528700501189480, −6.45339507635722220335713401436, −4.17480634615125495236651157008, −3.65219918801141344353284213698, −2.17068100688273565496093995118, 2.65651127756193653052569241611, 4.37454862741536426103971039111, 4.91272339051955037131901474020, 6.67442808011595797474835213534, 8.026378927783185830921300642317, 8.618812418898781115068009025307, 9.251907674416658519838077735070, 11.37440339637692220000359228979, 12.35861845346684169040724837887, 12.97525407287563767618946020011

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