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Properties

Label 2-252-7.2-c5-0-11
Degree $2$
Conductor $252$
Sign $0.415 + 0.909i$
Analytic cond. $40.4167$
Root an. cond. $6.35741$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (39.3 − 68.1i)5-s + (−100. − 81.7i)7-s + (345. + 598. i)11-s + 818.·13-s + (554. + 960. i)17-s + (286. − 496. i)19-s + (1.25e3 − 2.18e3i)23-s + (−1.53e3 − 2.65e3i)25-s + 3.25e3·29-s + (−5.05e3 − 8.76e3i)31-s + (−9.52e3 + 3.63e3i)35-s + (−2.43e3 + 4.21e3i)37-s + 1.30e4·41-s − 9.30e3·43-s + (6.45e3 − 1.11e4i)47-s + ⋯
L(s)  = 1  + (0.703 − 1.21i)5-s + (−0.776 − 0.630i)7-s + (0.861 + 1.49i)11-s + 1.34·13-s + (0.465 + 0.806i)17-s + (0.182 − 0.315i)19-s + (0.496 − 0.859i)23-s + (−0.490 − 0.849i)25-s + 0.719·29-s + (−0.945 − 1.63i)31-s + (−1.31 + 0.502i)35-s + (−0.292 + 0.506i)37-s + 1.21·41-s − 0.767·43-s + (0.426 − 0.738i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.415 + 0.909i$
Analytic conductor: \(40.4167\)
Root analytic conductor: \(6.35741\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :5/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.386432625\)
\(L(\frac12)\) \(\approx\) \(2.386432625\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7 \( 1 + (100. + 81.7i)T \)
good5 \( 1 + (-39.3 + 68.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-345. - 598. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 818.T + 3.71e5T^{2} \)
17 \( 1 + (-554. - 960. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-286. + 496. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-1.25e3 + 2.18e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 3.25e3T + 2.05e7T^{2} \)
31 \( 1 + (5.05e3 + 8.76e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (2.43e3 - 4.21e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.30e4T + 1.15e8T^{2} \)
43 \( 1 + 9.30e3T + 1.47e8T^{2} \)
47 \( 1 + (-6.45e3 + 1.11e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (9.77e3 + 1.69e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.25e4 + 2.17e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.56e4 + 2.71e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.79e4 + 4.84e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 2.05e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.38e4 - 5.85e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (7.03e3 - 1.21e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 7.71e4T + 3.93e9T^{2} \)
89 \( 1 + (-160. + 277. i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.12e5T + 8.58e9T^{2} \)
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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

−10.90368383188837416111849991405, −9.798839583721829296705283508013, −9.275095535148949458221282211382, −8.229222368643994247372610068220, −6.85594497469227791264481988558, −5.99793395320257582284441348414, −4.68624483818697190503454745326, −3.74032992166119261071068876800, −1.81624284247605151347362328761, −0.797144217960365651707493987760, 1.20334725628650198460127730390, 2.94167917997169344042337746507, 3.49225454350231285716834257931, 5.69348724733628945330041042892, 6.18934049461295855798113830056, 7.14590706211169204850096361106, 8.710756197050332085817890656746, 9.356806169626800297242103306230, 10.53606112551797535715733134737, 11.19385779710262615470566224633

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