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Properties

Label 2-252-12.11-c3-0-7
Degree $2$
Conductor $252$
Sign $0.973 - 0.229i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.973 - 0.229i$
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: \((2,\ 252,\ (\ :3/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.060739540\)
\(L(\frac12)\) \(\approx\) \(1.060739540\)
\(L(\frac{5}{2})\) 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

−11.54461971719040465564585438116, −10.61478782764226406598211262846, −9.884411256906405349005610652913, −8.917476775163298146669537061203, −7.88133080471272011356611171702, −6.89448444323433167646047756643, −5.15469773005828087763275242620, −4.03406239192333462767889696032, −2.73062751371421984885217073017, −1.22569918813528617210505539583, 0.55735370719762373170180062699, 2.64006324046183594242661076025, 4.68809339867224046960784806159, 5.35725575187955262657692385196, 6.92558490985855410549145816958, 7.30251522778971454305450321160, 8.796497048522980192156900713707, 9.335177945453991598225031432832, 10.36129912675028316463859598863, 11.51242952427921573955822222372

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