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Properties

Label 2-168-56.19-c1-0-13
Degree $2$
Conductor $168$
Sign $0.772 + 0.635i$
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  + (1.34 − 0.434i)2-s + (−0.866 + 0.5i)3-s + (1.62 − 1.16i)4-s + (1.44 − 2.49i)5-s + (−0.948 + 1.04i)6-s + (−2.63 + 0.194i)7-s + (1.67 − 2.27i)8-s + (0.499 − 0.866i)9-s + (0.856 − 3.98i)10-s + (2.91 + 5.04i)11-s + (−0.821 + 1.82i)12-s + 1.04·13-s + (−3.46 + 1.40i)14-s + 2.88i·15-s + (1.26 − 3.79i)16-s + (−5.91 + 3.41i)17-s + ⋯
L(s)  = 1  + (0.951 − 0.306i)2-s + (−0.499 + 0.288i)3-s + (0.811 − 0.584i)4-s + (0.644 − 1.11i)5-s + (−0.387 + 0.428i)6-s + (−0.997 + 0.0733i)7-s + (0.593 − 0.805i)8-s + (0.166 − 0.288i)9-s + (0.270 − 1.26i)10-s + (0.878 + 1.52i)11-s + (−0.237 + 0.526i)12-s + 0.290·13-s + (−0.926 + 0.375i)14-s + 0.744i·15-s + (0.317 − 0.948i)16-s + (−1.43 + 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.772 + 0.635i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.772 + 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62745 - 0.583286i\)
\(L(\frac12)\) \(\approx\) \(1.62745 - 0.583286i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 0.434i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.63 - 0.194i)T \)
good5 \( 1 + (-1.44 + 2.49i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.91 - 5.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.04T + 13T^{2} \)
17 \( 1 + (5.91 - 3.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.589 + 0.340i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.85 + 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.61iT - 29T^{2} \)
31 \( 1 + (1.91 + 3.31i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.06 + 1.19i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.19iT - 41T^{2} \)
43 \( 1 - 1.34T + 43T^{2} \)
47 \( 1 + (5.52 - 9.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.99 + 4.03i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.81 + 3.93i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.63 + 2.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.65 + 11.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.08iT - 71T^{2} \)
73 \( 1 + (-4.88 + 2.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.9 + 6.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.482iT - 83T^{2} \)
89 \( 1 + (-10.7 - 6.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.63iT - 97T^{2} \)
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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

−12.76757748002510149614795482892, −12.06331390313397351244067121741, −10.78622198854137575406187948322, −9.755170721669716510332174880221, −9.035260012394668252276566695807, −6.88025933215595418447040792340, −6.10045177864596035529977108331, −4.87676005778960035245180567586, −3.96907017172117565534504302796, −1.84146809126520651087391505517, 2.62291119429286503342540802073, 3.84652150762071490286595306586, 5.73068001940621656644904799721, 6.44094580446297855935906431891, 7.01279326763343115150200880704, 8.739854365737951880091495453333, 10.22840814143953579564661388607, 11.19971506497024468565252809712, 11.85148126908736759539171116531, 13.40768912663512112008077243502

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