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Properties

Label 2-168-168.107-c1-0-24
Degree $2$
Conductor $168$
Sign $0.910 + 0.413i$
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  + (1.18 − 0.765i)2-s + (0.996 + 1.41i)3-s + (0.826 − 1.82i)4-s + (1.00 − 1.73i)5-s + (2.27 + 0.920i)6-s + (−2.50 + 0.837i)7-s + (−0.412 − 2.79i)8-s + (−1.01 + 2.82i)9-s + (−0.137 − 2.82i)10-s + (−3.33 + 1.92i)11-s + (3.40 − 0.644i)12-s + 2.05i·13-s + (−2.34 + 2.91i)14-s + (3.45 − 0.310i)15-s + (−2.63 − 3.01i)16-s + (4.92 − 2.84i)17-s + ⋯
L(s)  = 1  + (0.840 − 0.541i)2-s + (0.575 + 0.817i)3-s + (0.413 − 0.910i)4-s + (0.447 − 0.775i)5-s + (0.926 + 0.375i)6-s + (−0.948 + 0.316i)7-s + (−0.145 − 0.989i)8-s + (−0.337 + 0.941i)9-s + (−0.0436 − 0.894i)10-s + (−1.00 + 0.580i)11-s + (0.982 − 0.186i)12-s + 0.571i·13-s + (−0.626 + 0.779i)14-s + (0.891 − 0.0802i)15-s + (−0.658 − 0.752i)16-s + (1.19 − 0.689i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.910 + 0.413i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.910 + 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90745 - 0.412490i\)
\(L(\frac12)\) \(\approx\) \(1.90745 - 0.412490i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 + 0.765i)T \)
3 \( 1 + (-0.996 - 1.41i)T \)
7 \( 1 + (2.50 - 0.837i)T \)
good5 \( 1 + (-1.00 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.33 - 1.92i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.05iT - 13T^{2} \)
17 \( 1 + (-4.92 + 2.84i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.232 - 0.403i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.711 - 1.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.25T + 29T^{2} \)
31 \( 1 + (-4.99 + 2.88i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.569 + 0.328i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.42iT - 41T^{2} \)
43 \( 1 - 6.77T + 43T^{2} \)
47 \( 1 + (1.79 - 3.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.50 + 2.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.72 - 3.30i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.50 - 4.91i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.78 + 6.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + (2.02 + 3.50i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.00488 - 0.00281i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.56iT - 83T^{2} \)
89 \( 1 + (-8.14 - 4.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.17T + 97T^{2} \)
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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

−12.90041894000396231381742164394, −11.94519550715430911882912581064, −10.61433102577120519283534938860, −9.670094970765548252994941384679, −9.228739817662974243580120661115, −7.50566113053300143653897252836, −5.75112749754389838599564997726, −4.97529806038942664366972522324, −3.63421524727235061388999145138, −2.33547852516916200294837164242, 2.69070554576602876990872728139, 3.50348987500091194742158683153, 5.68481660142153203651257371395, 6.44403727397346425697292066111, 7.49040457820174522601048947103, 8.311088590676453209183814886789, 9.879715062815210426477937225373, 10.99043197418865628161783852967, 12.44683451559942852261020071871, 13.01078286413004164027150722511

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