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Properties

Label 2-252-28.27-c3-0-8
Degree $2$
Conductor $252$
Sign $-1$
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  + 2.82i·2-s − 8.00·4-s + 3.74i·5-s + 18.5·7-s − 22.6i·8-s − 10.5·10-s + 43.8i·11-s + 52.3i·14-s + 64.0·16-s + 138. i·17-s − 153.·19-s − 29.9i·20-s − 124·22-s − 134. i·23-s + 111·25-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s + 0.334i·5-s + 0.999·7-s − 1.00i·8-s − 0.334·10-s + 1.20i·11-s + 1.00i·14-s + 1.00·16-s + 1.97i·17-s − 1.85·19-s − 0.334i·20-s − 1.20·22-s − 1.21i·23-s + 0.888·25-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.218523391\)
\(L(\frac12)\) \(\approx\) \(1.218523391\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 \)
7 \( 1 - 18.5T \)
good5 \( 1 - 3.74iT - 125T^{2} \)
11 \( 1 - 43.8iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 - 138. iT - 4.91e3T^{2} \)
19 \( 1 + 153.T + 6.85e3T^{2} \)
23 \( 1 + 134. iT - 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 + 201.T + 2.97e4T^{2} \)
37 \( 1 + 376T + 5.06e4T^{2} \)
41 \( 1 - 407. iT - 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 - 451. iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 - 1.55e3iT - 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−12.49120941426870261979493572963, −10.81600019966286131619037012763, −10.24398885802996811864389535766, −8.756601841402888717607626515084, −8.205480006563987035040820337681, −7.04623624308274785183467293402, −6.21373538213661619310916018758, −4.85943226521527646953576329997, −4.04269047118428318380418548279, −1.84415846645721254105135585550, 0.48709511171951585753736298159, 1.96565113700764387311927429282, 3.41617815242302051941430525025, 4.73489512190807667353403942285, 5.56187224883309104768616756545, 7.36052893459965553909994650489, 8.639732762113364179478430212485, 9.028160581419656138809257374940, 10.47434667707485202283485141093, 11.17391733774560011017745421471

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