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Properties

Label 2-252-7.5-c8-0-17
Degree $2$
Conductor $252$
Sign $0.997 - 0.0724i$
Analytic cond. $102.659$
Root an. cond. $10.1320$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (885. + 511. i)5-s + (765. − 2.27e3i)7-s + (89.3 + 154. i)11-s + 1.94e4i·13-s + (6.42e4 − 3.70e4i)17-s + (4.26e3 + 2.46e3i)19-s + (9.62e4 − 1.66e5i)23-s + (3.27e5 + 5.66e5i)25-s + 1.17e6·29-s + (−9.50e5 + 5.48e5i)31-s + (1.84e6 − 1.62e6i)35-s + (1.46e6 − 2.54e6i)37-s + 7.19e4i·41-s + 6.34e5·43-s + (1.90e5 + 1.09e5i)47-s + ⋯
L(s)  = 1  + (1.41 + 0.817i)5-s + (0.318 − 0.947i)7-s + (0.00610 + 0.0105i)11-s + 0.681i·13-s + (0.769 − 0.444i)17-s + (0.0327 + 0.0189i)19-s + (0.343 − 0.595i)23-s + (0.837 + 1.45i)25-s + 1.66·29-s + (−1.02 + 0.593i)31-s + (1.22 − 1.08i)35-s + (0.783 − 1.35i)37-s + 0.0254i·41-s + 0.185·43-s + (0.0389 + 0.0224i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0724i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.997 - 0.0724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.997 - 0.0724i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ 0.997 - 0.0724i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.422927146\)
\(L(\frac12)\) \(\approx\) \(3.422927146\)
\(L(5)\) 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 \( 1 \)
7 \( 1 + (-765. + 2.27e3i)T \)
good5 \( 1 + (-885. - 511. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-89.3 - 154. i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 1.94e4iT - 8.15e8T^{2} \)
17 \( 1 + (-6.42e4 + 3.70e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-4.26e3 - 2.46e3i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-9.62e4 + 1.66e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 1.17e6T + 5.00e11T^{2} \)
31 \( 1 + (9.50e5 - 5.48e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-1.46e6 + 2.54e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 7.19e4iT - 7.98e12T^{2} \)
43 \( 1 - 6.34e5T + 1.16e13T^{2} \)
47 \( 1 + (-1.90e5 - 1.09e5i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (6.01e5 + 1.04e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (8.57e6 - 4.95e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-1.33e7 - 7.70e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-9.22e5 - 1.59e6i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 3.59e7T + 6.45e14T^{2} \)
73 \( 1 + (-1.11e7 + 6.42e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (2.23e7 - 3.86e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 2.52e7iT - 2.25e15T^{2} \)
89 \( 1 + (-5.07e7 - 2.92e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 1.08e8iT - 7.83e15T^{2} \)
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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

−10.47882335356799685837450552025, −9.883480547681271997943793181447, −8.867187580553734949837507927329, −7.42704382000621444984394922157, −6.69855618250560423538699561262, −5.70367162947989190003905356268, −4.51428208668561477312863803572, −3.11897077393997262238399888178, −2.01457539779289533262614136782, −0.912428643462302275313990994659, 0.961181233616489548704671099272, 1.88931175818482460614873398964, 3.00544527366389042339214573477, 4.79858166982024903004594551662, 5.56714852963573423207073544044, 6.26410618908753000956676290707, 7.909752959838912904301467232966, 8.783969928431489656652388080057, 9.598973376669733784189538603251, 10.37133839917328755293166268379

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