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Properties

Label 2-252-63.5-c3-0-20
Degree $2$
Conductor $252$
Sign $-0.623 + 0.782i$
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  + (5.17 − 0.449i)3-s + (−6.39 − 11.0i)5-s + (−17.2 − 6.69i)7-s + (26.5 − 4.64i)9-s + (29.4 + 17.0i)11-s + (−8.83 − 5.09i)13-s + (−38.0 − 54.4i)15-s + (−55.3 − 95.8i)17-s + (−92.1 − 53.1i)19-s + (−92.3 − 26.8i)21-s + (−166. + 96.3i)23-s + (−19.2 + 33.2i)25-s + (135. − 36.0i)27-s + (59.2 − 34.2i)29-s − 296. i·31-s + ⋯
L(s)  = 1  + (0.996 − 0.0864i)3-s + (−0.571 − 0.990i)5-s + (−0.932 − 0.361i)7-s + (0.985 − 0.172i)9-s + (0.807 + 0.466i)11-s + (−0.188 − 0.108i)13-s + (−0.655 − 0.937i)15-s + (−0.789 − 1.36i)17-s + (−1.11 − 0.642i)19-s + (−0.960 − 0.279i)21-s + (−1.51 + 0.873i)23-s + (−0.153 + 0.266i)25-s + (0.966 − 0.256i)27-s + (0.379 − 0.219i)29-s − 1.71i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.782i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.623 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.623 + 0.782i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.623 + 0.782i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.488421320\)
\(L(\frac12)\) \(\approx\) \(1.488421320\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.17 + 0.449i)T \)
7 \( 1 + (17.2 + 6.69i)T \)
good5 \( 1 + (6.39 + 11.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-29.4 - 17.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (8.83 + 5.09i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (55.3 + 95.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (92.1 + 53.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (166. - 96.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-59.2 + 34.2i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 296. iT - 2.97e4T^{2} \)
37 \( 1 + (-58.0 + 100. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (92.4 - 160. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-77.9 - 135. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 218.T + 1.03e5T^{2} \)
53 \( 1 + (-478. + 275. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 313.T + 2.05e5T^{2} \)
61 \( 1 - 152. iT - 2.26e5T^{2} \)
67 \( 1 - 1.08e3T + 3.00e5T^{2} \)
71 \( 1 + 20.6iT - 3.57e5T^{2} \)
73 \( 1 + (-350. + 202. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 203.T + 4.93e5T^{2} \)
83 \( 1 + (-93.4 - 161. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (260. - 450. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-1.58e3 + 913. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.46059504671411723944964309398, −9.835294416058268517125056107261, −9.366327637039264501803941978015, −8.399184527509012606606822807256, −7.41637313923420647634623170777, −6.44126506743099475817362467878, −4.56164752304915167493910959567, −3.86606772903924960680141633558, −2.31064995431589651058683771419, −0.50008791487014396646718678459, 2.12945206250418940478987251375, 3.40372707088618006191040831159, 4.10497862015867953835587380976, 6.29742325213420525995785375220, 6.85064498623041125617049677687, 8.257768722679507894386982485050, 8.840085993990594976421032103340, 10.13958818105320233014200454809, 10.70368053119898330025885948935, 12.11962505231609693494372871583

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