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Properties

Label 2-168-168.11-c1-0-2
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: $\chi_{168} (11, \cdot )$
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

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

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