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Properties

Label 2-168-7.2-c3-0-4
Degree $2$
Conductor $168$
Sign $0.704 - 0.709i$
Analytic cond. $9.91232$
Root an. cond. $3.14838$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 2.59i)3-s + (−0.363 + 0.629i)5-s + (−18.1 + 3.72i)7-s + (−4.5 + 7.79i)9-s + (32.2 + 55.8i)11-s + 71.8·13-s + 2.17·15-s + (−24.4 − 42.3i)17-s + (−17.1 + 29.7i)19-s + (36.8 + 41.5i)21-s + (0.451 − 0.782i)23-s + (62.2 + 107. i)25-s + 27·27-s + 226.·29-s + (137. + 238. i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.0324 + 0.0562i)5-s + (−0.979 + 0.201i)7-s + (−0.166 + 0.288i)9-s + (0.883 + 1.53i)11-s + 1.53·13-s + 0.0375·15-s + (−0.348 − 0.604i)17-s + (−0.207 + 0.359i)19-s + (0.383 + 0.431i)21-s + (0.00409 − 0.00709i)23-s + (0.497 + 0.862i)25-s + 0.192·27-s + 1.45·29-s + (0.798 + 1.38i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ 0.704 - 0.709i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.22217 + 0.509124i\)
\(L(\frac12)\) \(\approx\) \(1.22217 + 0.509124i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 + (18.1 - 3.72i)T \)
good5 \( 1 + (0.363 - 0.629i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-32.2 - 55.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 71.8T + 2.19e3T^{2} \)
17 \( 1 + (24.4 + 42.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (17.1 - 29.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-0.451 + 0.782i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 226.T + 2.43e4T^{2} \)
31 \( 1 + (-137. - 238. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (147. - 255. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 186.T + 6.89e4T^{2} \)
43 \( 1 + 455.T + 7.95e4T^{2} \)
47 \( 1 + (141. - 244. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (178. + 308. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (364. + 631. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (137. - 237. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (96.6 + 167. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 40.5T + 3.57e5T^{2} \)
73 \( 1 + (-103. - 178. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (468. - 812. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 911.T + 5.71e5T^{2} \)
89 \( 1 + (-474. + 822. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 39.4T + 9.12e5T^{2} \)
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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

−12.43660727401934622297300893772, −11.68089834110410341377076343450, −10.44242461099817418339794653220, −9.435761652254825846933836028361, −8.386837347383658352001891858543, −6.81690283506879919776575017793, −6.44759044257803569557892629758, −4.78757480086060078791591204305, −3.25308796429087862763845695118, −1.42784863894874787918833459607, 0.71625979588820028325020477843, 3.23772157755120148675599749515, 4.19698869033753628681815854361, 6.06038810687165218885322778876, 6.44401024178718938475828505760, 8.410798336957025776810909991676, 9.062015822510530003132104627675, 10.36291841300224020834550250975, 11.08484122818576080341657500557, 12.08777828677733667711779428527

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