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Properties

Label 2-252-63.47-c1-0-4
Degree $2$
Conductor $252$
Sign $0.333 + 0.942i$
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  + (−1.44 − 0.956i)3-s + 2.86·5-s + (−1.83 − 1.90i)7-s + (1.16 + 2.76i)9-s − 2.71i·11-s + (3.18 − 1.84i)13-s + (−4.14 − 2.74i)15-s + (−3.22 − 5.58i)17-s + (2.73 + 1.58i)19-s + (0.819 + 4.50i)21-s − 2.99i·23-s + 3.22·25-s + (0.956 − 5.10i)27-s + (−2.48 − 1.43i)29-s + (8.26 + 4.77i)31-s + ⋯
L(s)  = 1  + (−0.833 − 0.552i)3-s + 1.28·5-s + (−0.692 − 0.721i)7-s + (0.389 + 0.920i)9-s − 0.817i·11-s + (0.884 − 0.510i)13-s + (−1.06 − 0.708i)15-s + (−0.781 − 1.35i)17-s + (0.628 + 0.362i)19-s + (0.178 + 0.983i)21-s − 0.623i·23-s + 0.645·25-s + (0.184 − 0.982i)27-s + (−0.461 − 0.266i)29-s + (1.48 + 0.857i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.333 + 0.942i$
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: \((2,\ 252,\ (\ :1/2),\ 0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895735 - 0.633002i\)
\(L(\frac12)\) \(\approx\) \(0.895735 - 0.633002i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 + 0.956i)T \)
7 \( 1 + (1.83 + 1.90i)T \)
good5 \( 1 - 2.86T + 5T^{2} \)
11 \( 1 + 2.71iT - 11T^{2} \)
13 \( 1 + (-3.18 + 1.84i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.22 + 5.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.73 - 1.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.99iT - 23T^{2} \)
29 \( 1 + (2.48 + 1.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.26 - 4.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.70 - 2.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.794 - 1.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.67 - 8.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.65 - 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.16 - 1.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.33 - 7.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.566 - 0.327i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.86 - 6.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 + (-11.0 + 6.39i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.59 + 4.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.92 + 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.14 - 5.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.2 - 7.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.82695141609553522328974245247, −10.82147713189595846595764862145, −10.14616503341330169342081196421, −9.123166896265861921226830650373, −7.73741561579912983269925195646, −6.47695228548924264582142928113, −6.04993006030177818650852659555, −4.81249755793926844035795945020, −2.91342194540165627773136029488, −1.05252846935472732246611356843, 1.97378115836210442940779006582, 3.83403524570532299934229589893, 5.26698114846211107486518284173, 6.08920872187191469956700076458, 6.78094676122083384320858498481, 8.750611949308542118242388861734, 9.558605874804241169145186151600, 10.17613314178145382939984657440, 11.20979091063230654891913959505, 12.22098052788197778068104064145

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