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Properties

Label 2-252-9.4-c3-0-9
Degree $2$
Conductor $252$
Sign $0.518 + 0.854i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.38 − 3.94i)3-s + (−8.26 + 14.3i)5-s + (−3.5 − 6.06i)7-s + (−4.10 + 26.6i)9-s + (−5.18 − 8.97i)11-s + (10.5 − 18.2i)13-s + (84.3 − 15.8i)15-s − 17.1·17-s + 121.·19-s + (−12.0 + 34.3i)21-s + (−8.97 + 15.5i)23-s + (−74.0 − 128. i)25-s + (119. − 74.0i)27-s + (−65.9 − 114. i)29-s + (155. − 269. i)31-s + ⋯
L(s)  = 1  + (−0.651 − 0.758i)3-s + (−0.738 + 1.27i)5-s + (−0.188 − 0.327i)7-s + (−0.152 + 0.988i)9-s + (−0.142 − 0.245i)11-s + (0.224 − 0.389i)13-s + (1.45 − 0.272i)15-s − 0.244·17-s + 1.46·19-s + (−0.125 + 0.356i)21-s + (−0.0814 + 0.140i)23-s + (−0.592 − 1.02i)25-s + (0.849 − 0.528i)27-s + (−0.422 − 0.731i)29-s + (0.902 − 1.56i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.518 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.518 + 0.854i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.518 + 0.854i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9871903719\)
\(L(\frac12)\) \(\approx\) \(0.9871903719\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.38 + 3.94i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good5 \( 1 + (8.26 - 14.3i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (5.18 + 8.97i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-10.5 + 18.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 17.1T + 4.91e3T^{2} \)
19 \( 1 - 121.T + 6.85e3T^{2} \)
23 \( 1 + (8.97 - 15.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (65.9 + 114. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-155. + 269. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 109.T + 5.06e4T^{2} \)
41 \( 1 + (-96.4 + 167. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (242. + 419. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-117. - 203. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 165.T + 1.48e5T^{2} \)
59 \( 1 + (-42.2 + 73.2i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-20.1 - 34.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (341. - 592. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.11e3T + 3.57e5T^{2} \)
73 \( 1 - 930.T + 3.89e5T^{2} \)
79 \( 1 + (177. + 306. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-729. - 1.26e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 816.T + 7.04e5T^{2} \)
97 \( 1 + (190. + 330. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.40913600422158074197776585132, −10.80209786537760757906546481660, −9.788660122188166818331018329996, −8.041721008826510170727471050785, −7.41851415206426524903654877173, −6.55401821305522352938767724280, −5.52093096401012609989240551889, −3.84571863442260155740541003354, −2.57335502205813977664624199817, −0.56415138432415960395807313646, 1.00017386803075655864145549608, 3.41751480734511802677142463449, 4.64373474179221638759283925037, 5.26665429148816150771585861840, 6.61141199103406838776950521455, 8.024757741122013420918535893680, 9.018742661685122317263068783113, 9.695565311026718440934651257007, 10.93308318107182613429490362245, 11.89632755800791781949307441635

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