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Properties

Label 2-168-7.5-c2-0-7
Degree $2$
Conductor $168$
Sign $-0.589 + 0.807i$
Analytic cond. $4.57766$
Root an. cond. $2.13954$
Motivic weight $2$
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 − 0.866i)3-s + (−4.68 − 2.70i)5-s + (−6.12 − 3.38i)7-s + (1.5 − 2.59i)9-s + (−5.26 − 9.12i)11-s − 12.0i·13-s − 9.36·15-s + (−20.8 + 12.0i)17-s + (30.9 + 17.8i)19-s + (−12.1 + 0.234i)21-s + (20.6 − 35.7i)23-s + (2.11 + 3.65i)25-s − 5.19i·27-s − 28.6·29-s + (−4.50 + 2.59i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−0.936 − 0.540i)5-s + (−0.875 − 0.483i)7-s + (0.166 − 0.288i)9-s + (−0.478 − 0.829i)11-s − 0.925i·13-s − 0.624·15-s + (−1.22 + 0.707i)17-s + (1.62 + 0.939i)19-s + (−0.577 + 0.0111i)21-s + (0.898 − 1.55i)23-s + (0.0844 + 0.146i)25-s − 0.192i·27-s − 0.988·29-s + (−0.145 + 0.0838i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.589 + 0.807i$
Analytic conductor: \(4.57766\)
Root analytic conductor: \(2.13954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1),\ -0.589 + 0.807i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.445548 - 0.877352i\)
\(L(\frac12)\) \(\approx\) \(0.445548 - 0.877352i\)
\(L(2)\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (6.12 + 3.38i)T \)
good5 \( 1 + (4.68 + 2.70i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (5.26 + 9.12i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 12.0iT - 169T^{2} \)
17 \( 1 + (20.8 - 12.0i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-30.9 - 17.8i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-20.6 + 35.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 28.6T + 841T^{2} \)
31 \( 1 + (4.50 - 2.59i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-1.90 + 3.30i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 9.20iT - 1.68e3T^{2} \)
43 \( 1 - 54.2T + 1.84e3T^{2} \)
47 \( 1 + (-14.3 - 8.27i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (9.58 + 16.5i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-10.8 + 6.25i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-84.1 - 48.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (0.499 + 0.864i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 14.8T + 5.04e3T^{2} \)
73 \( 1 + (51.0 - 29.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-50.7 + 87.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 22.7iT - 6.88e3T^{2} \)
89 \( 1 + (98.7 + 57.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 153. iT - 9.40e3T^{2} \)
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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

−12.54756719366440903848479630961, −11.21680424807017569627479513778, −10.23052538313009480450561631687, −8.927073639129785085529226473811, −8.098505900131915285248357441263, −7.16507819235212927907687155811, −5.76662888254332596781966825237, −4.10922078065697510283257994864, −3.02992347569008823257908954059, −0.56211963567798624412200170884, 2.61153502811589543929399640780, 3.76277017503891866087337415519, 5.14887514539449677988197920933, 6.97942906775804468715637002291, 7.48295114251052880975539001698, 9.177640950972331515151612100115, 9.540037190010227500391634348505, 11.12295558314292436880569364538, 11.71934736761625420456351754239, 13.02142145194601176423425751278

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