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Properties

Label 2-252-63.47-c3-0-6
Degree $2$
Conductor $252$
Sign $0.315 - 0.949i$
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  + (2.97 − 4.25i)3-s − 12.7·5-s + (2.83 + 18.3i)7-s + (−9.27 − 25.3i)9-s + 34.0i·11-s + (8.83 − 5.09i)13-s + (−38.0 + 54.4i)15-s + (55.3 + 95.8i)17-s + (−92.1 − 53.1i)19-s + (86.3 + 42.4i)21-s + 192. i·23-s + 38.4·25-s + (−135. − 36.0i)27-s + (59.2 + 34.2i)29-s + (256. + 148. i)31-s + ⋯
L(s)  = 1  + (0.572 − 0.819i)3-s − 1.14·5-s + (0.153 + 0.988i)7-s + (−0.343 − 0.939i)9-s + 0.932i·11-s + (0.188 − 0.108i)13-s + (−0.655 + 0.937i)15-s + (0.789 + 1.36i)17-s + (−1.11 − 0.642i)19-s + (0.897 + 0.440i)21-s + 1.74i·23-s + 0.307·25-s + (−0.966 − 0.256i)27-s + (0.379 + 0.219i)29-s + (1.48 + 0.858i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.315 - 0.949i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.315 - 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.254566417\)
\(L(\frac12)\) \(\approx\) \(1.254566417\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.97 + 4.25i)T \)
7 \( 1 + (-2.83 - 18.3i)T \)
good5 \( 1 + 12.7T + 125T^{2} \)
11 \( 1 - 34.0iT - 1.33e3T^{2} \)
13 \( 1 + (-8.83 + 5.09i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-55.3 - 95.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (92.1 + 53.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 192. iT - 1.21e4T^{2} \)
29 \( 1 + (-59.2 - 34.2i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-256. - 148. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-58.0 + 100. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-92.4 - 160. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-77.9 + 135. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (109. + 188. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (478. - 275. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (156. - 271. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-131. + 76.1i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (542. - 939. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 20.6iT - 3.57e5T^{2} \)
73 \( 1 + (-350. + 202. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-101. - 176. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (93.4 - 161. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-260. + 450. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.58e3 + 913. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.14003629076826397489366907593, −11.09412193517757417052033452473, −9.687320961346037441459439115236, −8.537162674103872414008859115669, −8.005102577853165877157761990871, −7.00813566052129650819371630558, −5.84248210813510593684908428849, −4.26194538186103682050870912925, −3.01342470887520171100960550326, −1.58447728602235284327324960155, 0.47619758630350101574215777753, 2.91546600013130165524244332403, 3.99826208065909423167587468479, 4.71853173835807356153059505259, 6.44252353457255439844351484184, 7.941570301118166566217896924767, 8.181612498485843693129983833545, 9.534338762268294681479116109830, 10.55917026936906456801675206547, 11.19108743312823048089012284687

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