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Properties

Label 2-252-252.23-c1-0-38
Degree $2$
Conductor $252$
Sign $0.483 + 0.875i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.801 − 1.16i)2-s + (1.41 + 1.00i)3-s + (−0.716 − 1.86i)4-s + (2.71 − 1.56i)5-s + (2.29 − 0.844i)6-s + (−2.39 − 1.11i)7-s + (−2.74 − 0.661i)8-s + (0.992 + 2.83i)9-s + (0.348 − 4.42i)10-s + (−1.84 + 3.19i)11-s + (0.858 − 3.35i)12-s + (0.398 − 0.691i)13-s + (−3.22 + 1.89i)14-s + (5.41 + 0.505i)15-s + (−2.97 + 2.67i)16-s + (−2.63 + 1.51i)17-s + ⋯
L(s)  = 1  + (0.566 − 0.824i)2-s + (0.815 + 0.578i)3-s + (−0.358 − 0.933i)4-s + (1.21 − 0.701i)5-s + (0.938 − 0.344i)6-s + (−0.906 − 0.423i)7-s + (−0.972 − 0.233i)8-s + (0.330 + 0.943i)9-s + (0.110 − 1.39i)10-s + (−0.556 + 0.964i)11-s + (0.247 − 0.968i)12-s + (0.110 − 0.191i)13-s + (−0.862 + 0.506i)14-s + (1.39 + 0.130i)15-s + (−0.743 + 0.668i)16-s + (−0.638 + 0.368i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.483 + 0.875i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82551 - 1.07734i\)
\(L(\frac12)\) \(\approx\) \(1.82551 - 1.07734i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.801 + 1.16i)T \)
3 \( 1 + (-1.41 - 1.00i)T \)
7 \( 1 + (2.39 + 1.11i)T \)
good5 \( 1 + (-2.71 + 1.56i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.84 - 3.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.398 + 0.691i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.63 - 1.51i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.92 - 2.26i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.10 + 1.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.16 + 4.71i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.16iT - 31T^{2} \)
37 \( 1 + (2.94 - 5.09i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (9.25 + 5.34i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.01 + 2.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + (6.12 - 3.53i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.509T + 59T^{2} \)
61 \( 1 - 9.56T + 61T^{2} \)
67 \( 1 + 7.45iT - 67T^{2} \)
71 \( 1 - 2.15T + 71T^{2} \)
73 \( 1 + (-1.27 - 2.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 + (-0.812 - 1.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.73 + 1.00i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.88 + 3.26i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.18597243433757538107752199686, −10.49656275021522798095704910671, −9.996246161407547413166187407049, −9.459551986992903680580719797484, −8.366743919904137265684237940111, −6.62354061585637332883966565854, −5.33827885056968628114878652131, −4.42952872327767742187843157450, −3.08374121384321666073034890077, −1.85512033813727450471355147883, 2.59909762298987464006194996486, 3.32299808522739766536757807322, 5.34286169973784115119212760107, 6.40775748974358245094530775960, 6.87763856115898313643882521056, 8.231177463336450464648908373461, 9.148815409766892080481356986033, 9.934781245096457586161164417544, 11.49668586665127716241546576124, 12.71358492244647992025815829208

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