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Properties

Label 2-168-168.101-c1-0-2
Degree $2$
Conductor $168$
Sign $-0.999 + 0.0295i$
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  + (−0.968 + 1.03i)2-s + (0.609 + 1.62i)3-s + (−0.124 − 1.99i)4-s + (−2.24 + 1.29i)5-s + (−2.26 − 0.942i)6-s + (−2.53 − 0.751i)7-s + (2.17 + 1.80i)8-s + (−2.25 + 1.97i)9-s + (0.839 − 3.57i)10-s + (−1.63 + 2.83i)11-s + (3.16 − 1.41i)12-s + 0.912·13-s + (3.23 − 1.88i)14-s + (−3.47 − 2.85i)15-s + (−3.96 + 0.495i)16-s + (2.39 − 4.14i)17-s + ⋯
L(s)  = 1  + (−0.684 + 0.728i)2-s + (0.351 + 0.936i)3-s + (−0.0621 − 0.998i)4-s + (−1.00 + 0.580i)5-s + (−0.923 − 0.384i)6-s + (−0.958 − 0.284i)7-s + (0.769 + 0.638i)8-s + (−0.752 + 0.658i)9-s + (0.265 − 1.13i)10-s + (−0.493 + 0.855i)11-s + (0.912 − 0.409i)12-s + 0.253·13-s + (0.863 − 0.504i)14-s + (−0.897 − 0.737i)15-s + (−0.992 + 0.123i)16-s + (0.580 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.999 + 0.0295i$
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: \((2,\ 168,\ (\ :1/2),\ -0.999 + 0.0295i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00767323 - 0.519742i\)
\(L(\frac12)\) \(\approx\) \(0.00767323 - 0.519742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.968 - 1.03i)T \)
3 \( 1 + (-0.609 - 1.62i)T \)
7 \( 1 + (2.53 + 0.751i)T \)
good5 \( 1 + (2.24 - 1.29i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.63 - 2.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.912T + 13T^{2} \)
17 \( 1 + (-2.39 + 4.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.66 - 4.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.45 - 2.57i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 + (-8.18 - 4.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.59 + 0.922i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.91T + 41T^{2} \)
43 \( 1 - 8.00iT - 43T^{2} \)
47 \( 1 + (-3.29 - 5.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.841 - 1.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.50 + 0.867i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.72 + 8.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.8 + 6.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.603iT - 71T^{2} \)
73 \( 1 + (1.29 + 0.746i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.0625 + 0.108i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.246iT - 83T^{2} \)
89 \( 1 + (-1.80 - 3.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.10iT - 97T^{2} \)
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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

−13.74522388332238388717934878934, −12.08610656098316032087557223769, −10.92538667633716026298339851573, −9.986459614602283287437007590405, −9.469810484113361836723879798334, −7.960615595283113889373411249895, −7.40862361593905690072920143491, −5.96488288590776610188766765438, −4.50611963779466687572668018959, −3.14766815219254795048625904515, 0.57842973314417070171856797140, 2.73321168783006184182023261993, 3.88922517985971844057853617941, 6.08980106806273318857476733163, 7.48000927669790005647426045961, 8.306013004913460933829713950293, 8.976150590441330350298765864167, 10.29035382156729841740754695600, 11.60582638270175162384257258722, 12.17128265866954224421942264232

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