Properties

Label 2-168-168.5-c1-0-23
Degree $2$
Conductor $168$
Sign $-0.981 - 0.191i$
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  + (−0.0857 − 1.41i)2-s + (−0.528 + 1.64i)3-s + (−1.98 + 0.242i)4-s + (−2.66 − 1.54i)5-s + (2.37 + 0.604i)6-s + (−1.46 − 2.20i)7-s + (0.511 + 2.78i)8-s + (−2.44 − 1.74i)9-s + (−1.94 + 3.89i)10-s + (0.621 + 1.07i)11-s + (0.650 − 3.40i)12-s − 5.98·13-s + (−2.98 + 2.25i)14-s + (3.95 − 3.58i)15-s + (3.88 − 0.960i)16-s + (−0.595 − 1.03i)17-s + ⋯
L(s)  = 1  + (−0.0606 − 0.998i)2-s + (−0.305 + 0.952i)3-s + (−0.992 + 0.121i)4-s + (−1.19 − 0.688i)5-s + (0.969 + 0.246i)6-s + (−0.552 − 0.833i)7-s + (0.180 + 0.983i)8-s + (−0.813 − 0.581i)9-s + (−0.615 + 1.23i)10-s + (0.187 + 0.324i)11-s + (0.187 − 0.982i)12-s − 1.66·13-s + (−0.798 + 0.602i)14-s + (1.02 − 0.926i)15-s + (0.970 − 0.240i)16-s + (−0.144 − 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.981 - 0.191i$
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: \((2,\ 168,\ (\ :1/2),\ -0.981 - 0.191i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0214386 + 0.221990i\)
\(L(\frac12)\) \(\approx\) \(0.0214386 + 0.221990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0857 + 1.41i)T \)
3 \( 1 + (0.528 - 1.64i)T \)
7 \( 1 + (1.46 + 2.20i)T \)
good5 \( 1 + (2.66 + 1.54i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.621 - 1.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.98T + 13T^{2} \)
17 \( 1 + (0.595 + 1.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.614 + 1.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.56 - 1.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.19T + 29T^{2} \)
31 \( 1 + (-1.33 + 0.773i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.334 + 0.193i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.44T + 41T^{2} \)
43 \( 1 + 8.29iT - 43T^{2} \)
47 \( 1 + (3.34 - 5.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.25 + 9.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.22 + 1.86i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.16 + 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.7 - 6.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.21iT - 71T^{2} \)
73 \( 1 + (8.92 - 5.15i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.41 - 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.22iT - 83T^{2} \)
89 \( 1 + (-6.94 + 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−12.03384903822452681761342717610, −11.39113439213581959178795933644, −10.21028578781072586695561495444, −9.585564281724804435994711780840, −8.483859296318725864888801852752, −7.20723516849391404115894213159, −4.99545867796691038143973107068, −4.34889365175874021826775475584, −3.21952884876913770188689905278, −0.21611288555846807522497819688, 3.05003783288679874829893440389, 4.89484806556720266290288042913, 6.23235860231301797372988878395, 7.06982031578713079564357082532, 7.86234427629538080914625420167, 8.859090092491976944593210943668, 10.26021057681136849594348847931, 11.73082946626220198263681322418, 12.30122135169566467170152536440, 13.31276780185500085030908130235

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