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Properties

Label 2-252-63.25-c3-0-18
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

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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: $\chi_{252} (25, \cdot )$
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 \( 1 + (666. - 1.15e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

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

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