↑

Properties

Label 2-252-63.58-c3-0-10
Degree $2$
Conductor $252$
Sign $0.301 - 0.953i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.31 + 2.90i)3-s + (−0.310 − 0.538i)5-s + (11.0 + 14.8i)7-s + (10.1 + 25.0i)9-s + (25.2 − 43.6i)11-s + (−31.3 + 54.2i)13-s + (0.222 − 3.22i)15-s + (15.7 + 27.3i)17-s + (−7.43 + 12.8i)19-s + (4.68 + 96.1i)21-s + (41.8 + 72.5i)23-s + (62.3 − 107. i)25-s + (−28.8 + 137. i)27-s + (88.3 + 152. i)29-s − 137.·31-s + ⋯
L(s)  = 1  + (0.829 + 0.558i)3-s + (−0.0278 − 0.0481i)5-s + (0.598 + 0.801i)7-s + (0.376 + 0.926i)9-s + (0.691 − 1.19i)11-s + (−0.668 + 1.15i)13-s + (0.00383 − 0.0554i)15-s + (0.225 + 0.390i)17-s + (−0.0898 + 0.155i)19-s + (0.0487 + 0.998i)21-s + (0.379 + 0.657i)23-s + (0.498 − 0.863i)25-s + (−0.205 + 0.978i)27-s + (0.565 + 0.979i)29-s − 0.795·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.301 - 0.953i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.301 - 0.953i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.510168406\)
\(L(\frac12)\) \(\approx\) \(2.510168406\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.31 - 2.90i)T \)
7 \( 1 + (-11.0 - 14.8i)T \)
good5 \( 1 + (0.310 + 0.538i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-25.2 + 43.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (31.3 - 54.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-15.7 - 27.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (7.43 - 12.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-41.8 - 72.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-88.3 - 152. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 137.T + 2.97e4T^{2} \)
37 \( 1 + (-9.32 + 16.1i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (36.7 - 63.6i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (6.41 + 11.1i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 111.T + 1.03e5T^{2} \)
53 \( 1 + (24.1 + 41.8i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 398.T + 2.05e5T^{2} \)
61 \( 1 - 357.T + 2.26e5T^{2} \)
67 \( 1 - 310.T + 3.00e5T^{2} \)
71 \( 1 - 968.T + 3.57e5T^{2} \)
73 \( 1 + (552. + 957. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 206.T + 4.93e5T^{2} \)
83 \( 1 + (381. + 660. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (465. - 806. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (666. + 1.15e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.67520404203809035152070123695, −10.88693802734720167755861119307, −9.640868580958637357083651806041, −8.830928894404808146955843356796, −8.247823979451490379775131906758, −6.88428433091859517707081182143, −5.49057911478783667688769941848, −4.36798190921631734000307004885, −3.13493534073155802376680885346, −1.75709816337729150566356944711, 0.985235287316364438218264677365, 2.41783652949468999013369915932, 3.84104777685196557773116997134, 5.03459016695819199380849824121, 6.80607635459871200988163261569, 7.42649049852052384074523220472, 8.310489786057274789414867462884, 9.500928422510215513298498774561, 10.27359438381060746839093422694, 11.52924215061265436731753981918

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