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Properties

Label 2-168-56.5-c2-0-18
Degree $2$
Conductor $168$
Sign $0.737 + 0.675i$
Analytic cond. $4.57766$
Root an. cond. $2.13954$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 − 1.49i)2-s + (0.866 + 1.5i)3-s + (−0.474 + 3.97i)4-s + (4.33 − 7.50i)5-s + (1.09 − 3.28i)6-s + (2.39 + 6.57i)7-s + (6.57 − 4.56i)8-s + (−1.5 + 2.59i)9-s + (−16.9 + 3.48i)10-s + (−9.70 + 5.60i)11-s + (−6.36 + 2.72i)12-s + 15.6·13-s + (6.65 − 12.3i)14-s + 15.0·15-s + (−15.5 − 3.76i)16-s + (29.3 − 16.9i)17-s + ⋯
L(s)  = 1  + (−0.663 − 0.747i)2-s + (0.288 + 0.5i)3-s + (−0.118 + 0.992i)4-s + (0.866 − 1.50i)5-s + (0.182 − 0.547i)6-s + (0.342 + 0.939i)7-s + (0.821 − 0.570i)8-s + (−0.166 + 0.288i)9-s + (−1.69 + 0.348i)10-s + (−0.882 + 0.509i)11-s + (−0.530 + 0.227i)12-s + 1.20·13-s + (0.475 − 0.879i)14-s + 1.00·15-s + (−0.971 − 0.235i)16-s + (1.72 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.737 + 0.675i$
Analytic conductor: \(4.57766\)
Root analytic conductor: \(2.13954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1),\ 0.737 + 0.675i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.29552 - 0.503406i\)
\(L(\frac12)\) \(\approx\) \(1.29552 - 0.503406i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 1.49i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
7 \( 1 + (-2.39 - 6.57i)T \)
good5 \( 1 + (-4.33 + 7.50i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (9.70 - 5.60i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 15.6T + 169T^{2} \)
17 \( 1 + (-29.3 + 16.9i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.81 + 11.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-0.498 + 0.864i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 13.0iT - 841T^{2} \)
31 \( 1 + (-6.91 + 3.99i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-11.3 - 6.55i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 0.655iT - 1.68e3T^{2} \)
43 \( 1 - 5.93iT - 1.84e3T^{2} \)
47 \( 1 + (21.8 + 12.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (57.5 - 33.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (12.4 + 21.5i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (47.8 - 82.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (29.6 - 17.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 61.8T + 5.04e3T^{2} \)
73 \( 1 + (28.0 - 16.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (23.6 - 40.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 117.T + 6.88e3T^{2} \)
89 \( 1 + (101. + 58.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 48.9iT - 9.40e3T^{2} \)
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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

−12.40844931182865767599148209472, −11.45605930321226202832237877009, −10.13055501183848963984167062625, −9.393189432417758179837544183232, −8.693932118849310228908152547754, −7.83390467513357687680372429154, −5.58318077566955316894150293736, −4.71774162836322609537559675231, −2.86314484347388308217765825966, −1.34922178573323271102134907137, 1.50474895998756827683049843647, 3.35414838708879950084619443152, 5.71578472180707768548854633681, 6.40349667999825192490115063193, 7.58303878372602815191306728289, 8.170000800306795156941145540877, 9.821684111808806045358848230474, 10.48095509436085518655605571562, 11.17261154691548699461222741643, 13.13642142356095078263501124994

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