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Properties

Label 2-252-63.41-c3-0-15
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: $\chi_{252} (41, \cdot )$
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

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

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