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Properties

Label 2-168-56.37-c1-0-10
Degree $2$
Conductor $168$
Sign $0.567 + 0.823i$
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  + (1.06 − 0.926i)2-s + (−0.866 + 0.5i)3-s + (0.283 − 1.97i)4-s + (1.23 + 0.710i)5-s + (−0.462 + 1.33i)6-s + (1.39 − 2.24i)7-s + (−1.53 − 2.37i)8-s + (0.499 − 0.866i)9-s + (1.97 − 0.380i)10-s + (0.832 − 0.480i)11-s + (0.744 + 1.85i)12-s + 3.57i·13-s + (−0.591 − 3.69i)14-s − 1.42·15-s + (−3.83 − 1.12i)16-s + (2.43 + 4.20i)17-s + ⋯
L(s)  = 1  + (0.755 − 0.655i)2-s + (−0.499 + 0.288i)3-s + (0.141 − 0.989i)4-s + (0.550 + 0.317i)5-s + (−0.188 + 0.545i)6-s + (0.527 − 0.849i)7-s + (−0.541 − 0.840i)8-s + (0.166 − 0.288i)9-s + (0.624 − 0.120i)10-s + (0.251 − 0.144i)11-s + (0.214 + 0.535i)12-s + 0.990i·13-s + (−0.158 − 0.987i)14-s − 0.366·15-s + (−0.959 − 0.280i)16-s + (0.589 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.567 + 0.823i$
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: \((2,\ 168,\ (\ :1/2),\ 0.567 + 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39486 - 0.732704i\)
\(L(\frac12)\) \(\approx\) \(1.39486 - 0.732704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.926i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-1.39 + 2.24i)T \)
good5 \( 1 + (-1.23 - 0.710i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.832 + 0.480i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.57iT - 13T^{2} \)
17 \( 1 + (-2.43 - 4.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.28 + 3.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.72 - 4.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.78iT - 29T^{2} \)
31 \( 1 + (-3.67 - 6.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.21 + 1.27i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.20T + 41T^{2} \)
43 \( 1 + 4.45iT - 43T^{2} \)
47 \( 1 + (-0.211 + 0.366i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.41 + 4.85i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.43 - 3.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.67 + 0.969i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.13 + 5.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.12T + 71T^{2} \)
73 \( 1 + (4.99 + 8.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.139 + 0.241i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.69iT - 83T^{2} \)
89 \( 1 + (1.07 - 1.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.44T + 97T^{2} \)
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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

−12.56938799573385822260296796761, −11.59794928609560572797844017696, −10.66938907174795992085567241283, −10.17404307825091949215976524546, −8.845250219924906789569927773915, −6.95895049091423565926152396937, −6.06226850533182893124367720123, −4.74978252213797943997544664611, −3.74389886061002782203515620520, −1.74584887351419321447524190920, 2.40801582588983615246315232067, 4.42291174415563444965057170276, 5.60503736468337330638990003906, 6.17931095861044432270347987797, 7.69378116255904908285601255340, 8.526735058395248902301919858452, 9.922471033530278224588784205069, 11.36465892091140496956468604458, 12.22295540045060281554045949821, 12.88619981005053379271428299957

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