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Properties

Label 2-252-12.11-c3-0-2
Degree $2$
Conductor $252$
Sign $-0.108 - 0.994i$
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  + (1.16 − 2.57i)2-s + (−5.29 − 5.99i)4-s − 0.300i·5-s + 7i·7-s + (−21.6 + 6.70i)8-s + (−0.774 − 0.348i)10-s − 9.71·11-s − 76.4·13-s + (18.0 + 8.13i)14-s + (−7.83 + 63.5i)16-s + 93.2i·17-s − 81.8i·19-s + (−1.79 + 1.59i)20-s + (−11.2 + 25.0i)22-s − 185.·23-s + ⋯
L(s)  = 1  + (0.410 − 0.911i)2-s + (−0.662 − 0.749i)4-s − 0.0268i·5-s + 0.377i·7-s + (−0.955 + 0.296i)8-s + (−0.0244 − 0.0110i)10-s − 0.266·11-s − 1.63·13-s + (0.344 + 0.155i)14-s + (−0.122 + 0.992i)16-s + 1.33i·17-s − 0.988i·19-s + (−0.0201 + 0.0177i)20-s + (−0.109 + 0.242i)22-s − 1.68·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.108 - 0.994i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.108 - 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2324562504\)
\(L(\frac12)\) \(\approx\) \(0.2324562504\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 + 2.57i)T \)
3 \( 1 \)
7 \( 1 - 7iT \)
good5 \( 1 + 0.300iT - 125T^{2} \)
11 \( 1 + 9.71T + 1.33e3T^{2} \)
13 \( 1 + 76.4T + 2.19e3T^{2} \)
17 \( 1 - 93.2iT - 4.91e3T^{2} \)
19 \( 1 + 81.8iT - 6.85e3T^{2} \)
23 \( 1 + 185.T + 1.21e4T^{2} \)
29 \( 1 - 159. iT - 2.43e4T^{2} \)
31 \( 1 - 139. iT - 2.97e4T^{2} \)
37 \( 1 - 30.5T + 5.06e4T^{2} \)
41 \( 1 - 173. iT - 6.89e4T^{2} \)
43 \( 1 + 356. iT - 7.95e4T^{2} \)
47 \( 1 + 346.T + 1.03e5T^{2} \)
53 \( 1 + 154. iT - 1.48e5T^{2} \)
59 \( 1 + 586.T + 2.05e5T^{2} \)
61 \( 1 - 167.T + 2.26e5T^{2} \)
67 \( 1 + 403. iT - 3.00e5T^{2} \)
71 \( 1 + 156.T + 3.57e5T^{2} \)
73 \( 1 + 565.T + 3.89e5T^{2} \)
79 \( 1 + 542. iT - 4.93e5T^{2} \)
83 \( 1 - 791.T + 5.71e5T^{2} \)
89 \( 1 - 809. iT - 7.04e5T^{2} \)
97 \( 1 + 192.T + 9.12e5T^{2} \)
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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

−12.11669334533293705555034891746, −10.87700838237436805632034794287, −10.14812358399586944179808977727, −9.197782250394126513212615936992, −8.167839833582037654563990666944, −6.68732283801298161686495868197, −5.40500755349160141559309246662, −4.50924547867944213530006686052, −3.06214168660453887132508695090, −1.88335361900822811555021115405, 0.07534335176835654972218015825, 2.68027056718725775483771897626, 4.19532352323972957342863876615, 5.14300609444083095339360853239, 6.29742197855594356105124888877, 7.44549795961077677090230818145, 7.981053944136325657056474826126, 9.431497510111434113193166299107, 10.09146290068617240695163962531, 11.69141452383343430014226676910

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