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Properties

Label 2-252-21.11-c8-0-15
Degree $2$
Conductor $252$
Sign $0.228 + 0.973i$
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  + (−318. + 183. i)5-s + (1.59e3 + 1.79e3i)7-s + (−9.68e3 − 5.59e3i)11-s + 3.90e4·13-s + (−7.82e4 − 4.51e4i)17-s + (−4.70e4 − 8.15e4i)19-s + (−1.77e5 + 1.02e5i)23-s + (−1.27e5 + 2.21e5i)25-s − 7.71e4i·29-s + (3.17e4 − 5.49e4i)31-s + (−8.36e5 − 2.78e5i)35-s + (2.89e5 + 5.02e5i)37-s − 1.45e6i·41-s + 7.68e4·43-s + (−4.67e5 + 2.70e5i)47-s + ⋯
L(s)  = 1  + (−0.509 + 0.293i)5-s + (0.663 + 0.748i)7-s + (−0.661 − 0.382i)11-s + 1.36·13-s + (−0.936 − 0.540i)17-s + (−0.361 − 0.625i)19-s + (−0.633 + 0.365i)23-s + (−0.327 + 0.566i)25-s − 0.109i·29-s + (0.0343 − 0.0594i)31-s + (−0.557 − 0.185i)35-s + (0.154 + 0.267i)37-s − 0.515i·41-s + 0.0224·43-s + (−0.0958 + 0.0553i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.228 + 0.973i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ 0.228 + 0.973i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.202987654\)
\(L(\frac12)\) \(\approx\) \(1.202987654\)
\(L(5)\) 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 \( 1 + (-1.59e3 - 1.79e3i)T \)
good5 \( 1 + (318. - 183. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (9.68e3 + 5.59e3i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 3.90e4T + 8.15e8T^{2} \)
17 \( 1 + (7.82e4 + 4.51e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (4.70e4 + 8.15e4i)T + (-8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.77e5 - 1.02e5i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 7.71e4iT - 5.00e11T^{2} \)
31 \( 1 + (-3.17e4 + 5.49e4i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-2.89e5 - 5.02e5i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 1.45e6iT - 7.98e12T^{2} \)
43 \( 1 - 7.68e4T + 1.16e13T^{2} \)
47 \( 1 + (4.67e5 - 2.70e5i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-1.13e7 - 6.56e6i)T + (3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (2.24e6 + 1.29e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (6.68e6 + 1.15e7i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (1.91e5 - 3.31e5i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 3.97e7iT - 6.45e14T^{2} \)
73 \( 1 + (-2.16e7 + 3.75e7i)T + (-4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (7.49e6 + 1.29e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + 1.06e7iT - 2.25e15T^{2} \)
89 \( 1 + (5.41e7 - 3.12e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 7.35e7T + 7.83e15T^{2} \)
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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

−10.76244274114892253573894474403, −9.239671569444155643933830067512, −8.460579744436760253991845052917, −7.61253814664243894565585030076, −6.35814236289717350242951102603, −5.36699674676599011892448239423, −4.18774380997224934758720974649, −2.97394482543559331568737866163, −1.80573044461477794175017291844, −0.29687848974167528102073444001, 0.961046838874239070844588767160, 2.12880692515887417481266791619, 3.83733450697071998234835083054, 4.44251637130516120977452688118, 5.79969164936999293311784262454, 6.94774026102119396706684804611, 8.132290405033269594329941159629, 8.519102162033582228491855552790, 10.07967874734566034383765604735, 10.82245136273500132100062420747

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