Properties

Label 2-252-63.20-c3-0-5
Degree $2$
Conductor $252$
Sign $-0.755 - 0.655i$
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  + (2.29 + 4.65i)3-s + (6.03 + 10.4i)5-s + (2.10 + 18.4i)7-s + (−16.4 + 21.4i)9-s + (0.00221 + 0.00127i)11-s + (6.06 − 3.50i)13-s + (−34.8 + 52.1i)15-s − 28.3·17-s − 49.1i·19-s + (−80.9 + 52.1i)21-s + (44.4 − 25.6i)23-s + (−10.3 + 17.9i)25-s + (−137. − 27.2i)27-s + (97.9 + 56.5i)29-s + (−28.4 + 16.4i)31-s + ⋯
L(s)  = 1  + (0.442 + 0.896i)3-s + (0.539 + 0.935i)5-s + (0.113 + 0.993i)7-s + (−0.608 + 0.793i)9-s + (6.06e−5 + 3.50e−5i)11-s + (0.129 − 0.0747i)13-s + (−0.599 + 0.897i)15-s − 0.403·17-s − 0.594i·19-s + (−0.840 + 0.541i)21-s + (0.403 − 0.232i)23-s + (−0.0828 + 0.143i)25-s + (−0.980 − 0.194i)27-s + (0.627 + 0.362i)29-s + (−0.164 + 0.0950i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.755 - 0.655i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.755 - 0.655i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.038879931\)
\(L(\frac12)\) \(\approx\) \(2.038879931\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.29 - 4.65i)T \)
7 \( 1 + (-2.10 - 18.4i)T \)
good5 \( 1 + (-6.03 - 10.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-0.00221 - 0.00127i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-6.06 + 3.50i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 28.3T + 4.91e3T^{2} \)
19 \( 1 + 49.1iT - 6.85e3T^{2} \)
23 \( 1 + (-44.4 + 25.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-97.9 - 56.5i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (28.4 - 16.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 101.T + 5.06e4T^{2} \)
41 \( 1 + (11.2 + 19.4i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (227. - 393. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-231. + 400. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 567. iT - 1.48e5T^{2} \)
59 \( 1 + (145. + 252. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-592. - 341. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-269. - 467. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 307. iT - 3.57e5T^{2} \)
73 \( 1 - 495. iT - 3.89e5T^{2} \)
79 \( 1 + (324. - 562. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-565. + 979. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 130.T + 7.04e5T^{2} \)
97 \( 1 + (-1.26e3 - 727. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.71766808499092398938543140369, −10.85121108204236385356098113569, −10.08329329331966687284717369068, −9.110495777923903787231037550062, −8.373054909816617354576018534213, −6.90588087847905545782642590320, −5.79326386050900889732246411460, −4.71289736386552112197708186430, −3.16224116393601060815406608509, −2.30093964153191064397005126692, 0.76839196468662343543506851452, 1.89535124712364907029204909402, 3.62493799975510319636284518445, 5.02512139178674247586607585855, 6.29983599878737392957540317951, 7.30598593872263220771100219080, 8.295249892917544649257410454737, 9.126485186836653846485743591346, 10.16417739072800440344824184738, 11.37584750998377132204074031484

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