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Properties

Label 2-252-28.27-c3-0-30
Degree $2$
Conductor $252$
Sign $0.492 - 0.870i$
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  + (2.78 + 0.501i)2-s + (7.49 + 2.79i)4-s − 4.57i·5-s + (−2.93 + 18.2i)7-s + (19.4 + 11.5i)8-s + (2.29 − 12.7i)10-s + 26.0i·11-s + 75.0i·13-s + (−17.3 + 49.4i)14-s + (48.4 + 41.8i)16-s − 115. i·17-s + 119.·19-s + (12.7 − 34.3i)20-s + (−13.0 + 72.4i)22-s + 61.4i·23-s + ⋯
L(s)  = 1  + (0.984 + 0.177i)2-s + (0.937 + 0.348i)4-s − 0.409i·5-s + (−0.158 + 0.987i)7-s + (0.860 + 0.509i)8-s + (0.0725 − 0.402i)10-s + 0.713i·11-s + 1.60i·13-s + (−0.330 + 0.943i)14-s + (0.756 + 0.654i)16-s − 1.65i·17-s + 1.43·19-s + (0.142 − 0.383i)20-s + (−0.126 + 0.701i)22-s + 0.556i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.492 - 0.870i$
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: \((2,\ 252,\ (\ :3/2),\ 0.492 - 0.870i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.368315556\)
\(L(\frac12)\) \(\approx\) \(3.368315556\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.78 - 0.501i)T \)
3 \( 1 \)
7 \( 1 + (2.93 - 18.2i)T \)
good5 \( 1 + 4.57iT - 125T^{2} \)
11 \( 1 - 26.0iT - 1.33e3T^{2} \)
13 \( 1 - 75.0iT - 2.19e3T^{2} \)
17 \( 1 + 115. iT - 4.91e3T^{2} \)
19 \( 1 - 119.T + 6.85e3T^{2} \)
23 \( 1 - 61.4iT - 1.21e4T^{2} \)
29 \( 1 - 71.9T + 2.43e4T^{2} \)
31 \( 1 + 231.T + 2.97e4T^{2} \)
37 \( 1 - 13.3T + 5.06e4T^{2} \)
41 \( 1 - 144. iT - 6.89e4T^{2} \)
43 \( 1 + 288. iT - 7.95e4T^{2} \)
47 \( 1 + 343.T + 1.03e5T^{2} \)
53 \( 1 + 142.T + 1.48e5T^{2} \)
59 \( 1 - 403.T + 2.05e5T^{2} \)
61 \( 1 - 21.7iT - 2.26e5T^{2} \)
67 \( 1 + 598. iT - 3.00e5T^{2} \)
71 \( 1 + 589. iT - 3.57e5T^{2} \)
73 \( 1 + 6.85iT - 3.89e5T^{2} \)
79 \( 1 + 972. iT - 4.93e5T^{2} \)
83 \( 1 + 429.T + 5.71e5T^{2} \)
89 \( 1 - 1.08e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.03e3iT - 9.12e5T^{2} \)
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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

−11.84636313241915028864876560575, −11.37939302543335336906616798764, −9.673261381916724369806003901297, −8.970209597274217497332187537650, −7.47631341968318743161709736960, −6.69608421029830065687391619906, −5.35543298041721641020600899665, −4.69240356892935975233159446137, −3.16745444676721680116023060105, −1.85461091448879896791086251068, 1.05507404476045346342373935930, 3.01327119371387730133722718769, 3.78363114998550111499775013707, 5.24564270942528477479286488303, 6.21346623209582606795481894192, 7.28237680490630261281198047008, 8.227276820840638408486380562617, 10.04439773259244162420963685265, 10.63251408212367809221602114922, 11.34783970696562806571059764114

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