Properties

Label 2-168-168.5-c1-0-6
Degree $2$
Conductor $168$
Sign $0.650 - 0.759i$
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  + (−0.368 + 1.36i)2-s + (−1.26 − 1.18i)3-s + (−1.72 − 1.00i)4-s + (1.54 + 0.894i)5-s + (2.08 − 1.29i)6-s + (2.63 + 0.230i)7-s + (2.00 − 1.99i)8-s + (0.206 + 2.99i)9-s + (−1.79 + 1.78i)10-s + (0.501 + 0.868i)11-s + (1.00 + 3.31i)12-s + 2.47·13-s + (−1.28 + 3.51i)14-s + (−0.904 − 2.96i)15-s + (1.97 + 3.47i)16-s + (3.32 + 5.76i)17-s + ⋯
L(s)  = 1  + (−0.260 + 0.965i)2-s + (−0.730 − 0.682i)3-s + (−0.864 − 0.502i)4-s + (0.692 + 0.399i)5-s + (0.849 − 0.528i)6-s + (0.996 + 0.0869i)7-s + (0.710 − 0.703i)8-s + (0.0686 + 0.997i)9-s + (−0.566 + 0.564i)10-s + (0.151 + 0.261i)11-s + (0.288 + 0.957i)12-s + 0.685·13-s + (−0.343 + 0.939i)14-s + (−0.233 − 0.765i)15-s + (0.494 + 0.869i)16-s + (0.807 + 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.650 - 0.759i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.650 - 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.845902 + 0.389384i\)
\(L(\frac12)\) \(\approx\) \(0.845902 + 0.389384i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.368 - 1.36i)T \)
3 \( 1 + (1.26 + 1.18i)T \)
7 \( 1 + (-2.63 - 0.230i)T \)
good5 \( 1 + (-1.54 - 0.894i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.501 - 0.868i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.47T + 13T^{2} \)
17 \( 1 + (-3.32 - 5.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.85 + 3.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.85 + 3.95i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.748T + 29T^{2} \)
31 \( 1 + (-2.87 + 1.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.22 + 1.86i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.01T + 41T^{2} \)
43 \( 1 - 9.19iT - 43T^{2} \)
47 \( 1 + (1.19 - 2.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.33 + 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.34 - 4.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.02 - 3.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.89 - 3.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.46iT - 71T^{2} \)
73 \( 1 + (5.68 - 3.28i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.53 + 4.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.65iT - 83T^{2} \)
89 \( 1 + (-7.39 + 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.75iT - 97T^{2} \)
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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

−13.13729057235427291174042218296, −12.00476121270336926694625014657, −10.77375963711578589155068673092, −10.00877531347115593558810681920, −8.447847576626659248205386013366, −7.71434399557879466739818334485, −6.40239096748176418390966532564, −5.82168197705163954869434291099, −4.53545579427287860850171755716, −1.63153078033791186569497763837, 1.40289546809316607456639871687, 3.57672627795837465257541343926, 4.91258667785655471552357243651, 5.74377701130514924892732715223, 7.76182442089430710756422499885, 8.998059161906758038909580213466, 9.827981081630063117698598092245, 10.65415931518280662764307726058, 11.71665966016874313801215584247, 12.11776652572312104226203195179

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