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Properties

Label 2-168-168.101-c1-0-22
Degree $2$
Conductor $168$
Sign $0.728 + 0.684i$
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  + (1.41 − 0.0117i)2-s + (−1.11 − 1.32i)3-s + (1.99 − 0.0332i)4-s + (−0.337 + 0.195i)5-s + (−1.58 − 1.86i)6-s + (1.39 − 2.24i)7-s + (2.82 − 0.0704i)8-s + (−0.529 + 2.95i)9-s + (−0.475 + 0.279i)10-s + (0.748 − 1.29i)11-s + (−2.26 − 2.61i)12-s − 3.28·13-s + (1.94 − 3.19i)14-s + (0.634 + 0.232i)15-s + (3.99 − 0.132i)16-s + (−1.68 + 2.91i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.00830i)2-s + (−0.641 − 0.766i)3-s + (0.999 − 0.0166i)4-s + (−0.151 + 0.0872i)5-s + (−0.648 − 0.761i)6-s + (0.527 − 0.849i)7-s + (0.999 − 0.0249i)8-s + (−0.176 + 0.984i)9-s + (−0.150 + 0.0884i)10-s + (0.225 − 0.390i)11-s + (−0.654 − 0.756i)12-s − 0.911·13-s + (0.520 − 0.853i)14-s + (0.163 + 0.0599i)15-s + (0.999 − 0.0332i)16-s + (−0.407 + 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.728 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57703 - 0.624710i\)
\(L(\frac12)\) \(\approx\) \(1.57703 - 0.624710i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0117i)T \)
3 \( 1 + (1.11 + 1.32i)T \)
7 \( 1 + (-1.39 + 2.24i)T \)
good5 \( 1 + (0.337 - 0.195i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.748 + 1.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.56 - 4.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.72 - 2.72i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.13T + 29T^{2} \)
31 \( 1 + (3.60 + 2.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.46 - 4.31i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 4.79iT - 43T^{2} \)
47 \( 1 + (2.51 + 4.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.499 + 0.864i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.36 + 0.785i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.40 + 5.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.05 + 1.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.3iT - 71T^{2} \)
73 \( 1 + (-2.76 - 1.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.239 + 0.414i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 + (-2.54 - 4.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.00iT - 97T^{2} \)
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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

−12.61025393047128968377327945967, −11.85753377345927147218093455675, −11.05081847281454659070499536849, −10.14112839999959431233131368362, −7.965810916189313798531876002816, −7.30737916436267096382336851323, −6.16464507838141891075128908054, −5.10342194965434514690534736074, −3.77483486049431270243480897101, −1.77018951701023219871796813763, 2.60695425756083414876482409530, 4.36599096439141978797933097640, 5.07202601810462057810433332766, 6.18231433625309635361914184732, 7.42499199789061893492461183477, 9.006846960282542141105334486238, 10.14726254872836216165279372123, 11.27220690847424386238473972267, 11.96347566574780229010466831885, 12.57992085714526902119741632431

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