Properties

Label 2-168-168.125-c1-0-20
Degree $2$
Conductor $168$
Sign $0.995 - 0.0980i$
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  + (1.22 + 0.707i)2-s + (1 − 1.41i)3-s + (0.999 + 1.73i)4-s − 1.41i·5-s + (2.22 − 1.02i)6-s + (−2 + 1.73i)7-s + 2.82i·8-s + (−1.00 − 2.82i)9-s + (1.00 − 1.73i)10-s + 2.44·11-s + (3.44 + 0.317i)12-s − 2·13-s + (−3.67 + 0.707i)14-s + (−2.00 − 1.41i)15-s + (−2.00 + 3.46i)16-s − 7.34·17-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.577 − 0.816i)3-s + (0.499 + 0.866i)4-s − 0.632i·5-s + (0.908 − 0.418i)6-s + (−0.755 + 0.654i)7-s + 0.999i·8-s + (−0.333 − 0.942i)9-s + (0.316 − 0.547i)10-s + 0.738·11-s + (0.995 + 0.0917i)12-s − 0.554·13-s + (−0.981 + 0.188i)14-s + (−0.516 − 0.365i)15-s + (−0.500 + 0.866i)16-s − 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.995 - 0.0980i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.995 - 0.0980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94751 + 0.0957434i\)
\(L(\frac12)\) \(\approx\) \(1.94751 + 0.0957434i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (-1 + 1.41i)T \)
7 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 7.34T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 + 4.89T + 29T^{2} \)
31 \( 1 - 6.92iT - 31T^{2} \)
37 \( 1 + 10.3iT - 37T^{2} \)
41 \( 1 - 2.44T + 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 5.65iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 - 1.41iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 - 7.34T + 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−12.80636514539274746955279783397, −12.33752560865530034600096942046, −11.33857314588446451715882709078, −9.228128192726993745966332214796, −8.731822711244444371767968159318, −7.30751051246633701970792744583, −6.53683517156225562797071974637, −5.32668073147634338698634020785, −3.75897831318437310597711497343, −2.33713640408367498360285531025, 2.57929940716044093213498160435, 3.71014308584302199155787165860, 4.68324490104985024585615052636, 6.30682349472081109387819837861, 7.29874322824967643759055039025, 9.163532332900421735715782390263, 9.925200374929835356106660199619, 10.83454839790621760549646616311, 11.61904246405206771521816083820, 13.09143946408156277135210290868

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