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Properties

Label 2-168-8.5-c1-0-1
Degree $2$
Conductor $168$
Sign $0.239 - 0.970i$
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  + (−1.40 − 0.114i)2-s + i·3-s + (1.97 + 0.321i)4-s + 1.12i·5-s + (0.114 − 1.40i)6-s + 7-s + (−2.74 − 0.678i)8-s − 9-s + (0.128 − 1.59i)10-s + 4.76i·11-s + (−0.321 + 1.97i)12-s − 0.456i·13-s + (−1.40 − 0.114i)14-s − 1.12·15-s + (3.79 + 1.26i)16-s + 0.415·17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0806i)2-s + 0.577i·3-s + (0.986 + 0.160i)4-s + 0.504i·5-s + (0.0465 − 0.575i)6-s + 0.377·7-s + (−0.970 − 0.239i)8-s − 0.333·9-s + (0.0407 − 0.503i)10-s + 1.43i·11-s + (−0.0928 + 0.569i)12-s − 0.126i·13-s + (−0.376 − 0.0304i)14-s − 0.291·15-s + (0.948 + 0.317i)16-s + 0.100·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.239 - 0.970i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.239 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.589178 + 0.461323i\)
\(L(\frac12)\) \(\approx\) \(0.589178 + 0.461323i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.114i)T \)
3 \( 1 - iT \)
7 \( 1 - T \)
good5 \( 1 - 1.12iT - 5T^{2} \)
11 \( 1 - 4.76iT - 11T^{2} \)
13 \( 1 + 0.456iT - 13T^{2} \)
17 \( 1 - 0.415T + 17T^{2} \)
19 \( 1 - 7.63iT - 19T^{2} \)
23 \( 1 - 1.58T + 23T^{2} \)
29 \( 1 + 6.72iT - 29T^{2} \)
31 \( 1 + 5.89T + 31T^{2} \)
37 \( 1 + 5.89iT - 37T^{2} \)
41 \( 1 + 0.415T + 41T^{2} \)
43 \( 1 + 9.43iT - 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 + 7.63iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 1.80iT - 61T^{2} \)
67 \( 1 - 8.09iT - 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 3.34T + 73T^{2} \)
79 \( 1 + 4.83T + 79T^{2} \)
83 \( 1 + 5.53iT - 83T^{2} \)
89 \( 1 - 4.92T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−12.62993963054990167497403141006, −11.79761677105948526614956417230, −10.65324354756101080060740548422, −10.09081089888238506646028816047, −9.113616535085354412377897211929, −7.902146556844121504462605962368, −7.01587039911524931431371448762, −5.59637651119471679744874324284, −3.85852569932768919364700249652, −2.14293046998215042750134426205, 1.03757062743597925431562949264, 2.91230890015459809053435706959, 5.21480186270970421957192463881, 6.47527668538168846022789089125, 7.52225015235297262632816231600, 8.669079058075056617435100333440, 9.105523707510394497537569588652, 10.80546537648266060398373876294, 11.30632029577931685680927672480, 12.44944610035311696257401732510

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