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Properties

Label 2-252-63.4-c1-0-3
Degree $2$
Conductor $252$
Sign $0.678 - 0.734i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
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  + 1.73i·3-s + 2·5-s + (2 − 1.73i)7-s − 2.99·9-s + 4·11-s + (−1.5 + 2.59i)13-s + 3.46i·15-s + (−3.5 + 6.06i)17-s + (−2.5 − 4.33i)19-s + (2.99 + 3.46i)21-s + 4·23-s − 25-s − 5.19i·27-s + (0.5 + 0.866i)29-s + (1.5 + 2.59i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + 0.894·5-s + (0.755 − 0.654i)7-s − 0.999·9-s + 1.20·11-s + (−0.416 + 0.720i)13-s + 0.894i·15-s + (−0.848 + 1.47i)17-s + (−0.573 − 0.993i)19-s + (0.654 + 0.755i)21-s + 0.834·23-s − 0.200·25-s − 0.999i·27-s + (0.0928 + 0.160i)29-s + (0.269 + 0.466i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36270 + 0.596231i\)
\(L(\frac12)\) \(\approx\) \(1.36270 + 0.596231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73iT \)
7 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 - 2T + 5T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.95764646429721083201520755685, −10.92292701491041295967004508239, −10.42828064234449101434286713912, −9.140408818947289994260450894603, −8.781905920340361212785597792389, −7.06579334550312997093215539801, −6.01452666127484331915187606485, −4.71891051159073265784383593327, −3.91098721878599581095103837476, −1.97229805884299553147153726157, 1.54870080446495394112970632088, 2.75271822423038963260300335811, 4.88020989592490665637607255954, 5.96273255125290601906640269212, 6.81126289867616638219070776844, 8.026776875535515146757411981750, 8.928448464120350357956573327813, 9.860824028105187945095381955488, 11.33106801869660453158138901078, 11.86587546310485270159398137294

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