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Properties

Label 2-252-63.47-c1-0-6
Degree $2$
Conductor $252$
Sign $0.932 + 0.360i$
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  + (1.71 − 0.242i)3-s + 0.699·5-s + (−0.461 − 2.60i)7-s + (2.88 − 0.831i)9-s + 0.265i·11-s + (−1.13 + 0.657i)13-s + (1.19 − 0.169i)15-s + (1.86 + 3.22i)17-s + (−0.382 − 0.220i)19-s + (−1.42 − 4.35i)21-s + 4.96i·23-s − 4.51·25-s + (4.74 − 2.12i)27-s + (−0.273 − 0.157i)29-s + (−4.85 − 2.80i)31-s + ⋯
L(s)  = 1  + (0.990 − 0.139i)3-s + 0.312·5-s + (−0.174 − 0.984i)7-s + (0.960 − 0.277i)9-s + 0.0799i·11-s + (−0.315 + 0.182i)13-s + (0.309 − 0.0437i)15-s + (0.452 + 0.783i)17-s + (−0.0877 − 0.0506i)19-s + (−0.310 − 0.950i)21-s + 1.03i·23-s − 0.902·25-s + (0.912 − 0.408i)27-s + (−0.0507 − 0.0292i)29-s + (−0.872 − 0.503i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70253 - 0.317610i\)
\(L(\frac12)\) \(\approx\) \(1.70253 - 0.317610i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.242i)T \)
7 \( 1 + (0.461 + 2.60i)T \)
good5 \( 1 - 0.699T + 5T^{2} \)
11 \( 1 - 0.265iT - 11T^{2} \)
13 \( 1 + (1.13 - 0.657i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.86 - 3.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.382 + 0.220i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.96iT - 23T^{2} \)
29 \( 1 + (0.273 + 0.157i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.85 + 2.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.351 - 0.608i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.39 - 9.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.73 + 6.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.50 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.51 - 4.91i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.73 - 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.89 - 2.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.97 + 5.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + (6.66 - 3.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.698 + 1.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.72 + 6.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.59 - 9.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.18 - 5.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.17665511555216738726054307013, −10.84183154583338124049887918080, −9.899924509038483175710267218232, −9.232321273203426155052492971199, −7.912522436581598100918057881040, −7.30442536163325761031226960680, −6.03446719646993826900028221426, −4.36697758501538061453183747636, −3.34298994217974885809136041587, −1.70651529942427102631710525937, 2.14947483260548795292350632890, 3.24347786379755255567568035890, 4.77351178451683699840502716878, 5.99757448155653306118654218955, 7.32693270880317344102744331651, 8.332478953354436467291261425652, 9.259079002794573426532610011391, 9.868393982308987740754873041213, 11.08640239766710917429329136218, 12.37692300634289287507610657690

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