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Properties

Label 2-252-63.25-c3-0-15
Degree $2$
Conductor $252$
Sign $-0.453 + 0.891i$
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  + (2.05 − 4.77i)3-s + (−7.73 + 13.3i)5-s + (10.1 + 15.4i)7-s + (−18.5 − 19.6i)9-s + (−34.4 − 59.7i)11-s + (−13.1 − 22.8i)13-s + (48.0 + 64.4i)15-s + (67.2 − 116. i)17-s + (−2.57 − 4.45i)19-s + (94.7 − 16.7i)21-s + (45.6 − 79.1i)23-s + (−57.1 − 99.0i)25-s + (−131. + 48.0i)27-s + (91.9 − 159. i)29-s − 98.5·31-s + ⋯
L(s)  = 1  + (0.395 − 0.918i)3-s + (−0.691 + 1.19i)5-s + (0.549 + 0.835i)7-s + (−0.686 − 0.727i)9-s + (−0.944 − 1.63i)11-s + (−0.280 − 0.486i)13-s + (0.826 + 1.10i)15-s + (0.960 − 1.66i)17-s + (−0.0310 − 0.0538i)19-s + (0.984 − 0.173i)21-s + (0.414 − 0.717i)23-s + (−0.457 − 0.792i)25-s + (−0.939 + 0.342i)27-s + (0.588 − 1.01i)29-s − 0.570·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.453 + 0.891i$
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: \((2,\ 252,\ (\ :3/2),\ -0.453 + 0.891i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.230046006\)
\(L(\frac12)\) \(\approx\) \(1.230046006\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.05 + 4.77i)T \)
7 \( 1 + (-10.1 - 15.4i)T \)
good5 \( 1 + (7.73 - 13.3i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (34.4 + 59.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (13.1 + 22.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-67.2 + 116. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.57 + 4.45i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-45.6 + 79.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-91.9 + 159. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 98.5T + 2.97e4T^{2} \)
37 \( 1 + (122. + 212. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-98.8 - 171. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (171. - 296. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 85.4T + 1.03e5T^{2} \)
53 \( 1 + (-97.7 + 169. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 416.T + 2.05e5T^{2} \)
61 \( 1 + 20.4T + 2.26e5T^{2} \)
67 \( 1 + 432.T + 3.00e5T^{2} \)
71 \( 1 + 623.T + 3.57e5T^{2} \)
73 \( 1 + (-105. + 183. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 1.23e3T + 4.93e5T^{2} \)
83 \( 1 + (-297. + 515. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-336. - 582. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-426. + 738. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.44069902874589738840440753200, −10.62141439381467917163917538967, −9.118015783062950947713438593932, −8.004315655415783004335713174721, −7.59661934494378525671802045752, −6.33527548705389471868103843048, −5.30736612587157986712418991440, −3.13063004220615449103837472973, −2.68551296481137889141583901554, −0.46272140321574892949445067176, 1.65891008543090643257748958103, 3.69618530333469665689577405855, 4.59998168099695530698845645757, 5.22770809821349115623778543410, 7.36167279675445296762162782857, 8.063364288334461294249841735121, 8.967788600754012783179553821296, 10.10735124485954538168128543914, 10.66935149175645074808939069383, 12.03164837938562574852977337957

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