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Properties

Label 2-168-8.5-c1-0-4
Degree $2$
Conductor $168$
Sign $0.992 - 0.121i$
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  + (1.19 − 0.756i)2-s + i·3-s + (0.856 − 1.80i)4-s + 4.10i·5-s + (0.756 + 1.19i)6-s + 7-s + (−0.343 − 2.80i)8-s − 9-s + (3.10 + 4.90i)10-s − 2.67i·11-s + (1.80 + 0.856i)12-s − 3.02i·13-s + (1.19 − 0.756i)14-s − 4.10·15-s + (−2.53 − 3.09i)16-s − 5.12·17-s + ⋯
L(s)  = 1  + (0.845 − 0.534i)2-s + 0.577i·3-s + (0.428 − 0.903i)4-s + 1.83i·5-s + (0.308 + 0.487i)6-s + 0.377·7-s + (−0.121 − 0.992i)8-s − 0.333·9-s + (0.981 + 1.55i)10-s − 0.807i·11-s + (0.521 + 0.247i)12-s − 0.838i·13-s + (0.319 − 0.202i)14-s − 1.05·15-s + (−0.633 − 0.773i)16-s − 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.992 - 0.121i$
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: \((2,\ 168,\ (\ :1/2),\ 0.992 - 0.121i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75768 + 0.107192i\)
\(L(\frac12)\) \(\approx\) \(1.75768 + 0.107192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.756i)T \)
3 \( 1 - iT \)
7 \( 1 - T \)
good5 \( 1 - 4.10iT - 5T^{2} \)
11 \( 1 + 2.67iT - 11T^{2} \)
13 \( 1 + 3.02iT - 13T^{2} \)
17 \( 1 + 5.12T + 17T^{2} \)
19 \( 1 + 2.78iT - 19T^{2} \)
23 \( 1 - 7.12T + 23T^{2} \)
29 \( 1 - 8.83iT - 29T^{2} \)
31 \( 1 + 1.42T + 31T^{2} \)
37 \( 1 + 1.42iT - 37T^{2} \)
41 \( 1 - 5.12T + 41T^{2} \)
43 \( 1 + 2.39iT - 43T^{2} \)
47 \( 1 + 9.56T + 47T^{2} \)
53 \( 1 - 2.78iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 5.17iT - 61T^{2} \)
67 \( 1 - 0.244iT - 67T^{2} \)
71 \( 1 + 4.27T + 71T^{2} \)
73 \( 1 + 4.15T + 73T^{2} \)
79 \( 1 - 6.25T + 79T^{2} \)
83 \( 1 - 9.35iT - 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 - 6.69T + 97T^{2} \)
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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

−12.96469110797673485585997647486, −11.34987956860681282065895596181, −10.98451095457600095203818093318, −10.38676571614856005567172501133, −9.013402375454997787180091614050, −7.23568005325900047598719943798, −6.29628129727940572141124963117, −5.04967786297854292469162408271, −3.50451381756908344567578205162, −2.68275868538750246289004239567, 1.91184017365368862940752300597, 4.30665514206911784230827900037, 4.97824684458435886029360903309, 6.28276572866599524089765463043, 7.53166793441920669013715216149, 8.488778980249549280099455526162, 9.328340791274012825436172790513, 11.38515124320642213491992681606, 12.08842355729510033553162983564, 13.00932823495184399991630996761

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