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Properties

Label 2-168-168.107-c1-0-14
Degree $2$
Conductor $168$
Sign $0.623 + 0.782i$
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  + (0.0689 − 1.41i)2-s + (0.728 + 1.57i)3-s + (−1.99 − 0.194i)4-s + (1.00 − 1.73i)5-s + (2.27 − 0.920i)6-s + (2.50 − 0.837i)7-s + (−0.412 + 2.79i)8-s + (−1.93 + 2.28i)9-s + (−2.37 − 1.53i)10-s + (3.33 − 1.92i)11-s + (−1.14 − 3.26i)12-s − 2.05i·13-s + (−1.00 − 3.60i)14-s + (3.45 + 0.310i)15-s + (3.92 + 0.775i)16-s + (−4.92 + 2.84i)17-s + ⋯
L(s)  = 1  + (0.0487 − 0.998i)2-s + (0.420 + 0.907i)3-s + (−0.995 − 0.0973i)4-s + (0.447 − 0.775i)5-s + (0.926 − 0.375i)6-s + (0.948 − 0.316i)7-s + (−0.145 + 0.989i)8-s + (−0.646 + 0.762i)9-s + (−0.752 − 0.484i)10-s + (1.00 − 0.580i)11-s + (−0.330 − 0.943i)12-s − 0.571i·13-s + (−0.269 − 0.962i)14-s + (0.891 + 0.0802i)15-s + (0.981 + 0.193i)16-s + (−1.19 + 0.689i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.623 + 0.782i$
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: \((2,\ 168,\ (\ :1/2),\ 0.623 + 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21883 - 0.587198i\)
\(L(\frac12)\) \(\approx\) \(1.21883 - 0.587198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0689 + 1.41i)T \)
3 \( 1 + (-0.728 - 1.57i)T \)
7 \( 1 + (-2.50 + 0.837i)T \)
good5 \( 1 + (-1.00 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.33 + 1.92i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.05iT - 13T^{2} \)
17 \( 1 + (4.92 - 2.84i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.232 - 0.403i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.711 - 1.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.25T + 29T^{2} \)
31 \( 1 + (4.99 - 2.88i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.569 - 0.328i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.42iT - 41T^{2} \)
43 \( 1 - 6.77T + 43T^{2} \)
47 \( 1 + (1.79 - 3.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.50 + 2.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.72 + 3.30i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.50 + 4.91i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.78 + 6.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + (2.02 + 3.50i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.00488 + 0.00281i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.56iT - 83T^{2} \)
89 \( 1 + (8.14 + 4.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.17T + 97T^{2} \)
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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

−12.71304216945701399201505337828, −11.24117816131923962033300431179, −10.91282588368287922760603233530, −9.551468649473672327538735755644, −8.922521596058705994782740144395, −8.060264074821226932104908294763, −5.65764858462865057605091029265, −4.62734867011210901583202360310, −3.63508761227651265514927402610, −1.74832448184525958769794280039, 2.10726045148231320893132985378, 4.15694190844097891925159393111, 5.76141948181943555037045259649, 6.84322998440855193410607557378, 7.41645216081794887181714616271, 8.781710458579367965737270817623, 9.379033878580322788815918580611, 11.09870775837198768916435107639, 12.13621982110016406598507203715, 13.26906879885149665419083379715

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