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Properties

Label 2-252-28.27-c1-0-13
Degree $2$
Conductor $252$
Sign $0.997 - 0.0636i$
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  + (1.28 + 0.599i)2-s + (1.28 + 1.53i)4-s − 3.33i·5-s + (1.56 − 2.13i)7-s + (0.719 + 2.73i)8-s + (2 − 4.27i)10-s + 0.936i·11-s + 1.87i·13-s + (3.28 − 1.79i)14-s + (−0.719 + 3.93i)16-s + 5.20i·17-s − 7.12·19-s + (5.12 − 4.27i)20-s + (−0.561 + 1.19i)22-s − 0.936i·23-s + ⋯
L(s)  = 1  + (0.905 + 0.424i)2-s + (0.640 + 0.768i)4-s − 1.49i·5-s + (0.590 − 0.807i)7-s + (0.254 + 0.967i)8-s + (0.632 − 1.35i)10-s + 0.282i·11-s + 0.519i·13-s + (0.876 − 0.480i)14-s + (−0.179 + 0.983i)16-s + 1.26i·17-s − 1.63·19-s + (1.14 − 0.955i)20-s + (−0.119 + 0.255i)22-s − 0.195i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.997 - 0.0636i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.997 - 0.0636i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10767 + 0.0671339i\)
\(L(\frac12)\) \(\approx\) \(2.10767 + 0.0671339i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.599i)T \)
3 \( 1 \)
7 \( 1 + (-1.56 + 2.13i)T \)
good5 \( 1 + 3.33iT - 5T^{2} \)
11 \( 1 - 0.936iT - 11T^{2} \)
13 \( 1 - 1.87iT - 13T^{2} \)
17 \( 1 - 5.20iT - 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
23 \( 1 + 0.936iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 - 1.46iT - 41T^{2} \)
43 \( 1 + 9.06iT - 43T^{2} \)
47 \( 1 - 6.24T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 4.79iT - 61T^{2} \)
67 \( 1 - 10.9iT - 67T^{2} \)
71 \( 1 + 3.86iT - 71T^{2} \)
73 \( 1 - 6.67iT - 73T^{2} \)
79 \( 1 - 2.39iT - 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 1.46iT - 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−12.46704801461461855416227475157, −11.33941317751467734211930500345, −10.33337500052156982074739456645, −8.716570320669932295560392336736, −8.199201998403981279907580734221, −6.96528112629200188798302820996, −5.77868922247834442174551932415, −4.54999947629504821640751486100, −4.10920113436614441052851173917, −1.79586169949253477839784317123, 2.30709495294607646192895702078, 3.17823182755974289963550278791, 4.69304626545211694618803372859, 5.92388708571680527569796532004, 6.73505616909807984622477097375, 7.910688366249232273735482437063, 9.417233150321645699042486356432, 10.59436952651132398717593587287, 11.10933999593914076469006296074, 11.92779331716208192570375138447

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