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Properties

Label 2-168-168.107-c1-0-2
Degree $2$
Conductor $168$
Sign $-0.508 - 0.860i$
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  + (−0.935 + 1.06i)2-s + (−1.45 − 0.945i)3-s + (−0.248 − 1.98i)4-s + (−0.187 + 0.325i)5-s + (2.36 − 0.654i)6-s + (0.198 + 2.63i)7-s + (2.33 + 1.59i)8-s + (1.21 + 2.74i)9-s + (−0.169 − 0.503i)10-s + (−4.18 + 2.41i)11-s + (−1.51 + 3.11i)12-s + 4.40i·13-s + (−2.98 − 2.25i)14-s + (0.580 − 0.294i)15-s + (−3.87 + 0.987i)16-s + (2.39 − 1.38i)17-s + ⋯
L(s)  = 1  + (−0.661 + 0.749i)2-s + (−0.837 − 0.545i)3-s + (−0.124 − 0.992i)4-s + (−0.0839 + 0.145i)5-s + (0.963 − 0.267i)6-s + (0.0750 + 0.997i)7-s + (0.826 + 0.563i)8-s + (0.404 + 0.914i)9-s + (−0.0535 − 0.159i)10-s + (−1.26 + 0.728i)11-s + (−0.437 + 0.899i)12-s + 1.22i·13-s + (−0.797 − 0.603i)14-s + (0.149 − 0.0760i)15-s + (−0.969 + 0.246i)16-s + (0.580 − 0.335i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.508 - 0.860i$
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: \((2,\ 168,\ (\ :1/2),\ -0.508 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.234867 + 0.411616i\)
\(L(\frac12)\) \(\approx\) \(0.234867 + 0.411616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.935 - 1.06i)T \)
3 \( 1 + (1.45 + 0.945i)T \)
7 \( 1 + (-0.198 - 2.63i)T \)
good5 \( 1 + (0.187 - 0.325i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.18 - 2.41i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.40iT - 13T^{2} \)
17 \( 1 + (-2.39 + 1.38i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.83 + 3.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.60 - 6.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.968T + 29T^{2} \)
31 \( 1 + (2.78 - 1.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.459 - 0.265i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.88iT - 41T^{2} \)
43 \( 1 - 0.747T + 43T^{2} \)
47 \( 1 + (-5.20 + 9.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.40 - 2.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.65 + 0.953i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.43 + 4.86i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.85 - 6.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.55T + 71T^{2} \)
73 \( 1 + (0.445 + 0.772i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (11.3 + 6.54i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + (4.70 + 2.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.31T + 97T^{2} \)
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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

−13.13823301638665461120278487873, −11.95683541826524478471898047525, −11.17069026960567972722052630928, −10.00982092640674650864663353496, −9.022679733688648290106389977744, −7.66455072591847451545163477708, −7.02315215729899169299022339916, −5.68595409688448603947502992008, −4.99588961161982166453387576050, −2.00511181118279206999719310976, 0.61013113914193895250526873961, 3.22257826915192446537133726618, 4.49711115898133916083308940861, 5.89746771559245658473205079463, 7.56484206070449166958167232356, 8.365539884473215440973484870088, 9.993546311231442379527111043396, 10.43827824236708258330146077287, 11.10392158630178208552609167779, 12.38551209143415995743996246097

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