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Properties

Label 2-252-7.3-c8-0-9
Degree $2$
Conductor $252$
Sign $0.696 - 0.717i$
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  + (133. − 77.3i)5-s + (−2.23e3 − 877. i)7-s + (−3.95e3 + 6.84e3i)11-s − 4.17e3i·13-s + (−1.10e5 − 6.38e4i)17-s + (1.39e5 − 8.03e4i)19-s + (1.64e4 + 2.84e4i)23-s + (−1.83e5 + 3.17e5i)25-s − 6.59e5·29-s + (−5.16e3 − 2.98e3i)31-s + (−3.67e5 + 5.52e4i)35-s + (3.29e5 + 5.70e5i)37-s − 1.10e6i·41-s + 4.57e6·43-s + (1.35e6 − 7.84e5i)47-s + ⋯
L(s)  = 1  + (0.214 − 0.123i)5-s + (−0.930 − 0.365i)7-s + (−0.269 + 0.467i)11-s − 0.146i·13-s + (−1.32 − 0.764i)17-s + (1.06 − 0.616i)19-s + (0.0586 + 0.101i)23-s + (−0.469 + 0.813i)25-s − 0.932·29-s + (−0.00559 − 0.00322i)31-s + (−0.244 + 0.0368i)35-s + (0.175 + 0.304i)37-s − 0.390i·41-s + 1.33·43-s + (0.278 − 0.160i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ 0.696 - 0.717i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.265514560\)
\(L(\frac12)\) \(\approx\) \(1.265514560\)
\(L(5)\) 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 \( 1 + (2.23e3 + 877. i)T \)
good5 \( 1 + (-133. + 77.3i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (3.95e3 - 6.84e3i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + 4.17e3iT - 8.15e8T^{2} \)
17 \( 1 + (1.10e5 + 6.38e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (-1.39e5 + 8.03e4i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (-1.64e4 - 2.84e4i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + 6.59e5T + 5.00e11T^{2} \)
31 \( 1 + (5.16e3 + 2.98e3i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-3.29e5 - 5.70e5i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 1.10e6iT - 7.98e12T^{2} \)
43 \( 1 - 4.57e6T + 1.16e13T^{2} \)
47 \( 1 + (-1.35e6 + 7.84e5i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (4.31e6 - 7.46e6i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (1.85e7 + 1.07e7i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-2.16e7 + 1.25e7i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (9.65e6 - 1.67e7i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 9.50e6T + 6.45e14T^{2} \)
73 \( 1 + (-6.91e6 - 3.99e6i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (1.32e6 + 2.29e6i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 5.37e7iT - 2.25e15T^{2} \)
89 \( 1 + (-3.33e7 + 1.92e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 2.56e7iT - 7.83e15T^{2} \)
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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

−10.73063304383075018210933229212, −9.562314424115869695150112776515, −9.162905996994889316452077651464, −7.58761375899702784663872848717, −6.89470711475210590552785082032, −5.71663723018970810683694015724, −4.60928815804614522898865530967, −3.36416619541524908071124991961, −2.24449742402392849546168402491, −0.73785087815471620380136321541, 0.37243038467135028281951216894, 1.95674817507699207403755588209, 3.06843874032464523081167967059, 4.19034136234666608457813439661, 5.68442460753204632615197450311, 6.34283763245255591160741764861, 7.52019819826220690555533112850, 8.690404144015235067338493123339, 9.518461419261189731728590383619, 10.44165580998872261185614725525

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