Properties

Label 2-168-168.107-c1-0-18
Degree $2$
Conductor $168$
Sign $0.710 + 0.703i$
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  + (−0.450 + 1.34i)2-s + (−0.0929 − 1.72i)3-s + (−1.59 − 1.20i)4-s + (−0.187 + 0.325i)5-s + (2.36 + 0.654i)6-s + (−0.198 − 2.63i)7-s + (2.33 − 1.59i)8-s + (−2.98 + 0.321i)9-s + (−0.351 − 0.398i)10-s + (4.18 − 2.41i)11-s + (−1.94 + 2.86i)12-s − 4.40i·13-s + (3.62 + 0.922i)14-s + (0.580 + 0.294i)15-s + (1.08 + 3.85i)16-s + (−2.39 + 1.38i)17-s + ⋯
L(s)  = 1  + (−0.318 + 0.947i)2-s + (−0.0536 − 0.998i)3-s + (−0.797 − 0.603i)4-s + (−0.0839 + 0.145i)5-s + (0.963 + 0.267i)6-s + (−0.0750 − 0.997i)7-s + (0.826 − 0.563i)8-s + (−0.994 + 0.107i)9-s + (−0.111 − 0.125i)10-s + (1.26 − 0.728i)11-s + (−0.560 + 0.828i)12-s − 1.22i·13-s + (0.969 + 0.246i)14-s + (0.149 + 0.0760i)15-s + (0.270 + 0.962i)16-s + (−0.580 + 0.335i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.710 + 0.703i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782304 - 0.321736i\)
\(L(\frac12)\) \(\approx\) \(0.782304 - 0.321736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.450 - 1.34i)T \)
3 \( 1 + (0.0929 + 1.72i)T \)
7 \( 1 + (0.198 + 2.63i)T \)
good5 \( 1 + (0.187 - 0.325i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.18 + 2.41i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.40iT - 13T^{2} \)
17 \( 1 + (2.39 - 1.38i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.83 + 3.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.60 - 6.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.968T + 29T^{2} \)
31 \( 1 + (-2.78 + 1.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.459 + 0.265i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.88iT - 41T^{2} \)
43 \( 1 - 0.747T + 43T^{2} \)
47 \( 1 + (-5.20 + 9.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.40 - 2.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.65 - 0.953i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.43 - 4.86i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.85 - 6.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.55T + 71T^{2} \)
73 \( 1 + (0.445 + 0.772i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-11.3 - 6.54i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 + (-4.70 - 2.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.31T + 97T^{2} \)
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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

−13.16457490646207030150810055753, −11.64525788965685308463821215142, −10.66932292048187171283331080451, −9.351141849826412941146908711195, −8.263956574493538975934089681136, −7.35860287324166107744244167692, −6.55947578453129103424020214293, −5.49029928105455809227931437435, −3.68343914530422648913984860562, −0.981453426341582315034742916089, 2.28589264349944620694813826725, 3.94168818943820832641508493576, 4.79264951630804086200307463738, 6.44990847801676697606217747613, 8.387200754590129763396961822851, 9.175517677867360678367539461614, 9.763078121062560517755741813356, 10.95386335793450735548246783459, 12.00240445031495050273533835324, 12.26624176626045530058592222559

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