Properties

Label 2-168-168.5-c1-0-21
Degree $2$
Conductor $168$
Sign $0.948 + 0.317i$
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 + (0.0574 − 1.73i)3-s + (1.55 + 1.26i)4-s + (−0.461 − 0.266i)5-s + (0.896 − 2.27i)6-s + (0.489 − 2.60i)7-s + (1.46 + 2.41i)8-s + (−2.99 − 0.198i)9-s + (−0.488 − 0.573i)10-s + (2.28 + 3.96i)11-s + (2.27 − 2.61i)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 + (0.0331 − 0.999i)3-s + (0.775 + 0.631i)4-s + (−0.206 − 0.119i)5-s + (0.365 − 0.930i)6-s + (0.184 − 0.982i)7-s + (0.519 + 0.854i)8-s + (−0.997 − 0.0663i)9-s + (−0.154 − 0.181i)10-s + (0.689 + 1.19i)11-s + (0.656 − 0.754i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.948 + 0.317i$
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: \((2,\ 168,\ (\ :1/2),\ 0.948 + 0.317i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82432 - 0.297232i\)
\(L(\frac12)\) \(\approx\) \(1.82432 - 0.297232i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0574 + 1.73i)T \)
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

−12.69731195703255133415779959156, −12.16573157447482775251917867819, −11.19323878322202464542676426602, −9.793153791188198859371881620247, −8.038046725470234090646682439637, −7.33734286560560865912398106832, −6.57086185026125423611811337745, −5.08328954346981331756380218563, −3.84642348580121596924076589633, −2.01949235808062537425917585645, 2.67710893328147298392608783375, 3.81436150626982788833325761948, 5.20200419351759905443503252955, 5.87832746277666806934946545707, 7.57202348543211833616976800296, 9.146092891116976551989663158436, 9.865700987305922669996992893503, 11.38469649335195599261398020121, 11.52292165850485823891878327745, 12.72861495734043019558915299812

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