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Properties

Label 2-252-63.47-c3-0-2
Degree $2$
Conductor $252$
Sign $-0.921 + 0.388i$
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  + (−1.17 + 5.06i)3-s − 6.49·5-s + (18.4 − 2.01i)7-s + (−24.2 − 11.8i)9-s + 37.0i·11-s + (−39.5 + 22.8i)13-s + (7.62 − 32.8i)15-s + (55.2 + 95.7i)17-s + (−131. − 76.0i)19-s + (−11.3 + 95.5i)21-s − 123. i·23-s − 82.8·25-s + (88.6 − 108. i)27-s + (−89.2 − 51.5i)29-s + (−162. − 93.9i)31-s + ⋯
L(s)  = 1  + (−0.225 + 0.974i)3-s − 0.580·5-s + (0.994 − 0.109i)7-s + (−0.897 − 0.440i)9-s + 1.01i·11-s + (−0.844 + 0.487i)13-s + (0.131 − 0.565i)15-s + (0.788 + 1.36i)17-s + (−1.59 − 0.918i)19-s + (−0.118 + 0.992i)21-s − 1.12i·23-s − 0.662·25-s + (0.631 − 0.775i)27-s + (−0.571 − 0.329i)29-s + (−0.942 − 0.544i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.921 + 0.388i$
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: \((2,\ 252,\ (\ :3/2),\ -0.921 + 0.388i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4167810812\)
\(L(\frac12)\) \(\approx\) \(0.4167810812\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.17 - 5.06i)T \)
7 \( 1 + (-18.4 + 2.01i)T \)
good5 \( 1 + 6.49T + 125T^{2} \)
11 \( 1 - 37.0iT - 1.33e3T^{2} \)
13 \( 1 + (39.5 - 22.8i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-55.2 - 95.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (131. + 76.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 123. iT - 1.21e4T^{2} \)
29 \( 1 + (89.2 + 51.5i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (162. + 93.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-131. + 228. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (152. + 264. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (167. - 290. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-214. - 371. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (35.3 - 20.4i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (82.1 - 142. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (674. - 389. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (163. - 283. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 94.0iT - 3.57e5T^{2} \)
73 \( 1 + (334. - 193. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (245. + 424. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (621. - 1.07e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (560. - 971. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-1.01e3 - 587. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.00909071136980530196323301249, −11.03708618629874821779491308303, −10.40507361617582510920563183273, −9.305758252496864451469189605677, −8.313178472800501924396174624996, −7.32729447546005161400288822462, −5.89532633674724890100013752226, −4.53409608013149017916426740692, −4.13650927434778576572763861880, −2.17844102601572679729958908318, 0.16291554889231492060871668135, 1.73591662395694314516893314969, 3.28729804171139729431173906692, 5.00320976365104498643008222162, 5.90157536626642038628040213167, 7.34319159980147655209881561114, 7.906646927918653561146883924896, 8.783737163967242199631934866645, 10.32235496829384019295199486677, 11.46211783417410495719205547326

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