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Properties

Label 2-168-168.107-c1-0-11
Degree $2$
Conductor $168$
Sign $0.395 - 0.918i$
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 + 0.191i)2-s + (−0.402 + 1.68i)3-s + (1.92 + 0.535i)4-s + (−1.09 + 1.89i)5-s + (−0.885 + 2.28i)6-s + (−0.451 − 2.60i)7-s + (2.59 + 1.11i)8-s + (−2.67 − 1.35i)9-s + (−1.89 + 2.44i)10-s + (1.45 − 0.837i)11-s + (−1.67 + 3.03i)12-s + 1.56i·13-s + (−0.134 − 3.73i)14-s + (−2.74 − 2.60i)15-s + (3.42 + 2.06i)16-s + (0.278 − 0.160i)17-s + ⋯
L(s)  = 1  + (0.990 + 0.135i)2-s + (−0.232 + 0.972i)3-s + (0.963 + 0.267i)4-s + (−0.488 + 0.845i)5-s + (−0.361 + 0.932i)6-s + (−0.170 − 0.985i)7-s + (0.918 + 0.395i)8-s + (−0.892 − 0.451i)9-s + (−0.598 + 0.772i)10-s + (0.437 − 0.252i)11-s + (−0.484 + 0.874i)12-s + 0.432i·13-s + (−0.0360 − 0.999i)14-s + (−0.709 − 0.671i)15-s + (0.856 + 0.516i)16-s + (0.0676 − 0.0390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.395 - 0.918i$
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: \((2,\ 168,\ (\ :1/2),\ 0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45653 + 0.959047i\)
\(L(\frac12)\) \(\approx\) \(1.45653 + 0.959047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.191i)T \)
3 \( 1 + (0.402 - 1.68i)T \)
7 \( 1 + (0.451 + 2.60i)T \)
good5 \( 1 + (1.09 - 1.89i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.45 + 0.837i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.56iT - 13T^{2} \)
17 \( 1 + (-0.278 + 0.160i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.87 + 4.97i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.26 + 5.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.04T + 29T^{2} \)
31 \( 1 + (7.76 - 4.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.946 - 0.546i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.44iT - 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + (-2.25 + 3.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.42 - 7.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.37 - 3.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.80 - 2.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.716 - 1.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.37T + 71T^{2} \)
73 \( 1 + (4.49 + 7.77i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.58 + 2.07i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.06iT - 83T^{2} \)
89 \( 1 + (4.57 + 2.64i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.23T + 97T^{2} \)
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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

−13.14275991954084740906920234417, −11.75787399366649414489782472946, −11.05116592983981857303164176455, −10.45838752032306352061329430322, −9.020738524003265287516787231425, −7.32296257187850218388168493594, −6.62601097867078014453682212653, −5.14728407830632598499313225233, −3.98606445212444858744221266463, −3.15072433605012407573384517628, 1.72113171638745305228338638945, 3.46075597906306470943616687023, 5.19722568882828574525322371525, 5.88859802990027800224732942236, 7.25308347574856696733796112814, 8.216016338615096555887725727406, 9.565534010002671072008306916315, 11.30536238777556549238133062937, 11.86412043293082415574776676038, 12.71001045306385638920234601330

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