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Properties

Label 2-168-56.37-c1-0-6
Degree $2$
Conductor $168$
Sign $0.710 - 0.704i$
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  + (1.36 + 0.378i)2-s + (−0.866 + 0.5i)3-s + (1.71 + 1.03i)4-s + (−0.586 − 0.338i)5-s + (−1.36 + 0.353i)6-s + (2.23 + 1.41i)7-s + (1.94 + 2.05i)8-s + (0.499 − 0.866i)9-s + (−0.670 − 0.683i)10-s + (−1.44 + 0.835i)11-s + (−1.99 − 0.0372i)12-s − 1.28i·13-s + (2.51 + 2.77i)14-s + 0.677·15-s + (1.86 + 3.53i)16-s + (−3.18 − 5.51i)17-s + ⋯
L(s)  = 1  + (0.963 + 0.267i)2-s + (−0.499 + 0.288i)3-s + (0.856 + 0.516i)4-s + (−0.262 − 0.151i)5-s + (−0.559 + 0.144i)6-s + (0.845 + 0.534i)7-s + (0.687 + 0.726i)8-s + (0.166 − 0.288i)9-s + (−0.212 − 0.216i)10-s + (−0.436 + 0.251i)11-s + (−0.577 − 0.0107i)12-s − 0.355i·13-s + (0.671 + 0.740i)14-s + 0.174·15-s + (0.467 + 0.884i)16-s + (−0.771 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.710 - 0.704i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.710 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58147 + 0.651037i\)
\(L(\frac12)\) \(\approx\) \(1.58147 + 0.651037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.378i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-2.23 - 1.41i)T \)
good5 \( 1 + (0.586 + 0.338i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.44 - 0.835i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.28iT - 13T^{2} \)
17 \( 1 + (3.18 + 5.51i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.20 + 1.27i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.127 + 0.221i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.27iT - 29T^{2} \)
31 \( 1 + (-2.14 - 3.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.62 - 3.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.43T + 41T^{2} \)
43 \( 1 + 5.48iT - 43T^{2} \)
47 \( 1 + (4.73 - 8.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.91 - 2.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.74 + 5.04i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-13.2 - 7.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.6 - 7.88i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.48T + 71T^{2} \)
73 \( 1 + (1.43 + 2.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.09 + 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.63iT - 83T^{2} \)
89 \( 1 + (3.40 - 5.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.477T + 97T^{2} \)
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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

−12.91047387235865795973036899777, −11.81780819786675920997023264618, −11.39487623894842030274601924970, −10.19792899924924259931888714533, −8.606718089868118682868997527929, −7.56461302192876551164364726508, −6.30868948499760051334442230539, −5.10659930076959644255770885796, −4.40662245830030807811798035504, −2.54260355066980171573933139323, 1.82510339186801720882403117497, 3.80866992738194842329078913366, 4.90717891496742138359044280378, 6.12046652038525332502149529650, 7.18756511063491092577208708576, 8.295579655620457729469527490440, 10.18749891798999624080205111085, 11.05066915036678291643565203793, 11.57498894120481762233548966000, 12.81264997098511460539761683386

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