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Properties

Label 2-252-63.25-c3-0-12
Degree $2$
Conductor $252$
Sign $0.959 + 0.281i$
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  + (−1.41 − 4.99i)3-s + (2.84 − 4.92i)5-s + (15.2 + 10.4i)7-s + (−22.9 + 14.1i)9-s + (29.7 + 51.5i)11-s + (45.3 + 78.5i)13-s + (−28.6 − 7.24i)15-s + (51.8 − 89.8i)17-s + (−54.7 − 94.9i)19-s + (30.5 − 91.2i)21-s + (−0.737 + 1.27i)23-s + (46.3 + 80.2i)25-s + (103. + 94.9i)27-s + (−17.8 + 30.8i)29-s − 12.4·31-s + ⋯
L(s)  = 1  + (−0.272 − 0.962i)3-s + (0.254 − 0.440i)5-s + (0.825 + 0.563i)7-s + (−0.851 + 0.523i)9-s + (0.816 + 1.41i)11-s + (0.967 + 1.67i)13-s + (−0.492 − 0.124i)15-s + (0.740 − 1.28i)17-s + (−0.661 − 1.14i)19-s + (0.317 − 0.948i)21-s + (−0.00669 + 0.0115i)23-s + (0.370 + 0.642i)25-s + (0.735 + 0.677i)27-s + (−0.114 + 0.197i)29-s − 0.0723·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.959 + 0.281i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.959 + 0.281i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.964717370\)
\(L(\frac12)\) \(\approx\) \(1.964717370\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.41 + 4.99i)T \)
7 \( 1 + (-15.2 - 10.4i)T \)
good5 \( 1 + (-2.84 + 4.92i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-29.7 - 51.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-45.3 - 78.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-51.8 + 89.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (54.7 + 94.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (0.737 - 1.27i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (17.8 - 30.8i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 12.4T + 2.97e4T^{2} \)
37 \( 1 + (-26.2 - 45.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (127. + 220. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-148. + 256. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 475.T + 1.03e5T^{2} \)
53 \( 1 + (106. - 185. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 823.T + 2.05e5T^{2} \)
61 \( 1 + 804.T + 2.26e5T^{2} \)
67 \( 1 - 178.T + 3.00e5T^{2} \)
71 \( 1 + 308.T + 3.57e5T^{2} \)
73 \( 1 + (94.1 - 163. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 525.T + 4.93e5T^{2} \)
83 \( 1 + (-557. + 965. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-296. - 514. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (491. - 850. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.82810382259430445073858939608, −10.94080516392242500937950216060, −9.206302451346218531133826746662, −8.856884629878174793210386100401, −7.36374338117204802699903454013, −6.71985816955344799755109589604, −5.40659206268525297778851417043, −4.41489793478651148957446380392, −2.22291429128490938068735039496, −1.29048381433216595299864562522, 1.01041681409019545928140576216, 3.28917727732282758696032170104, 4.07280719485330400055977458835, 5.70090230118596878955804690901, 6.15686587199730232538925298666, 8.094300417217675492312913096661, 8.549197858851768956483586601093, 10.11150560539854103035317146854, 10.67572459647130037814275056821, 11.23377723098659369631956337265

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