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Properties

Label 2-252-28.23-c2-0-4
Degree $2$
Conductor $252$
Sign $-0.895 - 0.444i$
Analytic cond. $6.86650$
Root an. cond. $2.62040$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + (−2.5 + 4.33i)5-s + (3.5 + 6.06i)7-s − 8·8-s + (5 − 8.66i)10-s + (4.5 − 2.59i)11-s − 16·13-s + (−7 − 12.1i)14-s + 16·16-s + (2 + 3.46i)17-s + (−24 − 13.8i)19-s + (−10 + 17.3i)20-s + (−9 + 5.19i)22-s + (18 + 10.3i)23-s + ⋯
L(s)  = 1  − 2-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 8-s + (0.5 − 0.866i)10-s + (0.409 − 0.236i)11-s − 1.23·13-s + (−0.5 − 0.866i)14-s + 16-s + (0.117 + 0.203i)17-s + (−1.26 − 0.729i)19-s + (−0.5 + 0.866i)20-s + (−0.409 + 0.236i)22-s + (0.782 + 0.451i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(6.86650\)
Root analytic conductor: \(2.62040\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.110005 + 0.469585i\)
\(L(\frac12)\) \(\approx\) \(0.110005 + 0.469585i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 \)
7 \( 1 + (-3.5 - 6.06i)T \)
good5 \( 1 + (2.5 - 4.33i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-4.5 + 2.59i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 16T + 169T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (24 + 13.8i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-18 - 10.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 37T + 841T^{2} \)
31 \( 1 + (16.5 - 9.52i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (34 - 58.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 40T + 1.68e3T^{2} \)
43 \( 1 + 24.2iT - 1.84e3T^{2} \)
47 \( 1 + (39 + 22.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-36.5 - 63.2i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (25.5 - 14.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-41 + 71.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-69 + 39.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 72.7iT - 5.04e3T^{2} \)
73 \( 1 + (-47 - 81.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (7.5 + 4.33i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 133. iT - 6.88e3T^{2} \)
89 \( 1 + (-29 + 50.2i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 47T + 9.40e3T^{2} \)
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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

−11.79906783323975043595711834243, −11.24782058228467908874940718660, −10.35058234618629675778958434970, −9.229204363297385230215660796513, −8.432590409958215447889125327091, −7.33869403458961801077387015627, −6.60445915150119787401413871639, −5.19393073762215121705779764625, −3.27252605890380604013731236205, −2.04140708130326546469473324192, 0.32895776080592676130603360073, 1.90392431763012690063887888982, 3.91154425744722703592841949945, 5.15024043401435518413688639119, 6.79262546723181356585555742788, 7.62581454123998463521237943246, 8.470909475489330255895962738702, 9.412962910366325222918024549965, 10.38942135252758136537686831724, 11.26316531550431873327048451304

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