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Properties

Label 2-168-168.5-c1-0-17
Degree $2$
Conductor $168$
Sign $-0.703 + 0.710i$
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  + (−1.33 − 0.473i)2-s + (−1.52 − 0.815i)3-s + (1.55 + 1.26i)4-s + (0.461 + 0.266i)5-s + (1.64 + 1.81i)6-s + (0.489 − 2.60i)7-s + (−1.46 − 2.41i)8-s + (1.66 + 2.49i)9-s + (−0.488 − 0.573i)10-s + (−2.28 − 3.96i)11-s + (−1.34 − 3.19i)12-s − 4.97·13-s + (−1.88 + 3.23i)14-s + (−0.487 − 0.783i)15-s + (0.814 + 3.91i)16-s + (−2.16 − 3.74i)17-s + ⋯
L(s)  = 1  + (−0.942 − 0.334i)2-s + (−0.882 − 0.470i)3-s + (0.775 + 0.631i)4-s + (0.206 + 0.119i)5-s + (0.673 + 0.739i)6-s + (0.184 − 0.982i)7-s + (−0.519 − 0.854i)8-s + (0.556 + 0.830i)9-s + (−0.154 − 0.181i)10-s + (−0.689 − 1.19i)11-s + (−0.387 − 0.922i)12-s − 1.38·13-s + (−0.503 + 0.864i)14-s + (−0.125 − 0.202i)15-s + (0.203 + 0.979i)16-s + (−0.524 − 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.703 + 0.710i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169729 - 0.406794i\)
\(L(\frac12)\) \(\approx\) \(0.169729 - 0.406794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.473i)T \)
3 \( 1 + (1.52 + 0.815i)T \)
7 \( 1 + (-0.489 + 2.60i)T \)
good5 \( 1 + (-0.461 - 0.266i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.28 + 3.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.97T + 13T^{2} \)
17 \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.921 + 1.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.103 - 0.0596i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.74T + 29T^{2} \)
31 \( 1 + (1.93 - 1.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.02 - 4.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 + 1.87iT - 43T^{2} \)
47 \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.29 - 3.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.71 - 3.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.07 + 1.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.4 + 6.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.20iT - 71T^{2} \)
73 \( 1 + (-8.35 + 4.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0228 - 0.0396i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.86iT - 83T^{2} \)
89 \( 1 + (-8.23 + 14.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.18iT - 97T^{2} \)
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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

−12.12414132134298303612483020888, −11.24939775291043702887618540435, −10.53069813569659034936103461184, −9.675474867345385624159981773782, −8.110560347706080844136837687958, −7.30841903312090178329806883578, −6.33131628196812524173837994788, −4.76035605569545081118976148093, −2.64066637439857087712299548598, −0.59448241111334721536112248539, 2.18790094616905768109592915771, 4.82665992752518276108499435404, 5.70552512988038768510010015669, 6.89299096086578909927068368240, 8.051072824690882080670911334653, 9.440213201970919758471319273334, 9.934354499911401879343770657360, 10.99071800135869969884471444228, 12.04425826584458029715520860085, 12.69666809518076976517169284068

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