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Properties

Label 2-252-9.7-c3-0-5
Degree $2$
Conductor $252$
Sign $0.424 - 0.905i$
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  + (3.16 − 4.11i)3-s + (8.76 + 15.1i)5-s + (−3.5 + 6.06i)7-s + (−6.92 − 26.0i)9-s + (−26.5 + 45.9i)11-s + (45.5 + 78.8i)13-s + (90.3 + 11.9i)15-s − 86.1·17-s − 22.5·19-s + (13.8 + 33.6i)21-s + (−21.7 − 37.7i)23-s + (−91.1 + 157. i)25-s + (−129. − 54.1i)27-s + (60.4 − 104. i)29-s + (16.0 + 27.8i)31-s + ⋯
L(s)  = 1  + (0.609 − 0.792i)3-s + (0.784 + 1.35i)5-s + (−0.188 + 0.327i)7-s + (−0.256 − 0.966i)9-s + (−0.727 + 1.26i)11-s + (0.970 + 1.68i)13-s + (1.55 + 0.206i)15-s − 1.22·17-s − 0.272·19-s + (0.144 + 0.349i)21-s + (−0.197 − 0.341i)23-s + (−0.729 + 1.26i)25-s + (−0.922 − 0.385i)27-s + (0.387 − 0.670i)29-s + (0.0930 + 0.161i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.424 - 0.905i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.424 - 0.905i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.151393317\)
\(L(\frac12)\) \(\approx\) \(2.151393317\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-3.16 + 4.11i)T \)
7 \( 1 + (3.5 - 6.06i)T \)
good5 \( 1 + (-8.76 - 15.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (26.5 - 45.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-45.5 - 78.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 86.1T + 4.91e3T^{2} \)
19 \( 1 + 22.5T + 6.85e3T^{2} \)
23 \( 1 + (21.7 + 37.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-60.4 + 104. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-16.0 - 27.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 304.T + 5.06e4T^{2} \)
41 \( 1 + (-250. - 433. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-245. + 425. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (147. - 255. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 391.T + 1.48e5T^{2} \)
59 \( 1 + (105. + 183. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-122. + 211. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-10.7 - 18.6i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 87.0T + 3.57e5T^{2} \)
73 \( 1 - 45.0T + 3.89e5T^{2} \)
79 \( 1 + (-411. + 713. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-385. + 667. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 136.T + 7.04e5T^{2} \)
97 \( 1 + (-731. + 1.26e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.75697530316133938110403842938, −10.84500381421092394911258241110, −9.726560782045352981992286380279, −8.973159148919926502572847699908, −7.65863772044239226742123394785, −6.60637003662355999013993030605, −6.27861449440585050775638240904, −4.23855400522520871517984393028, −2.58751680308312912187384958209, −1.99217610123056717037780796812, 0.78037691721293480205106026155, 2.66637795288782728008023617495, 4.00057101809156605967717365563, 5.25208349605913711710434540990, 5.93767802442472200359182724977, 7.992890540549799950834825951468, 8.589838234429774489496035977136, 9.357521663211075519638734459406, 10.47356861003214735102329143223, 11.03733750332704428235582203674

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