Properties

Label 2-252-63.20-c3-0-16
Degree $2$
Conductor $252$
Sign $0.979 + 0.202i$
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  + (5.19 − 0.205i)3-s + (−2.34 − 4.05i)5-s + (18.4 + 1.99i)7-s + (26.9 − 2.13i)9-s + (16.1 + 9.30i)11-s + (−44.1 + 25.4i)13-s + (−12.9 − 20.5i)15-s + 112.·17-s − 111. i·19-s + (96.0 + 6.59i)21-s + (−124. + 71.8i)23-s + (51.5 − 89.2i)25-s + (139. − 16.6i)27-s + (206. + 119. i)29-s + (179. − 103. i)31-s + ⋯
L(s)  = 1  + (0.999 − 0.0395i)3-s + (−0.209 − 0.362i)5-s + (0.994 + 0.107i)7-s + (0.996 − 0.0790i)9-s + (0.441 + 0.255i)11-s + (−0.941 + 0.543i)13-s + (−0.223 − 0.354i)15-s + 1.60·17-s − 1.34i·19-s + (0.997 + 0.0684i)21-s + (−1.12 + 0.651i)23-s + (0.412 − 0.713i)25-s + (0.992 − 0.118i)27-s + (1.32 + 0.764i)29-s + (1.04 − 0.602i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.979 + 0.202i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.979 + 0.202i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.839890343\)
\(L(\frac12)\) \(\approx\) \(2.839890343\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.19 + 0.205i)T \)
7 \( 1 + (-18.4 - 1.99i)T \)
good5 \( 1 + (2.34 + 4.05i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-16.1 - 9.30i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (44.1 - 25.4i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 112.T + 4.91e3T^{2} \)
19 \( 1 + 111. iT - 6.85e3T^{2} \)
23 \( 1 + (124. - 71.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-206. - 119. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-179. + 103. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 227.T + 5.06e4T^{2} \)
41 \( 1 + (133. + 230. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (170. - 294. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-111. + 193. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 547. iT - 1.48e5T^{2} \)
59 \( 1 + (-43.9 - 76.1i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-312. - 180. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (372. + 645. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 135. iT - 3.57e5T^{2} \)
73 \( 1 + 467. iT - 3.89e5T^{2} \)
79 \( 1 + (-192. + 332. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (597. - 1.03e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.38e3T + 7.04e5T^{2} \)
97 \( 1 + (1.07e3 + 617. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.99266846669650312447106408547, −10.41113346450780794766099851721, −9.530157895092848232484501339230, −8.564149196913262238236602169841, −7.80663261651313204886367687278, −6.86196637376092961450191790826, −5.07666822105941503282742802558, −4.20333797961964795601811425760, −2.68122875805149137760345032968, −1.32120513795024763265004922819, 1.42375993030583185075921489841, 2.91078537872527203928751511951, 4.04751692144242671987984708436, 5.33003959712902620187825481831, 6.89576493608210795686250146695, 8.031422291417596238094389300819, 8.312891909543986689276446773645, 9.959056499943745985863177964103, 10.29917100362441008984856770576, 11.85793551374673847996656555100

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