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Properties

Label 2-252-252.103-c1-0-11
Degree $2$
Conductor $252$
Sign $0.986 - 0.163i$
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  + (−1.05 − 0.943i)2-s + (1.67 − 0.427i)3-s + (0.217 + 1.98i)4-s + (−2.49 + 1.44i)5-s + (−2.17 − 1.13i)6-s + (0.0846 + 2.64i)7-s + (1.64 − 2.29i)8-s + (2.63 − 1.43i)9-s + (3.99 + 0.839i)10-s + (1.99 + 1.15i)11-s + (1.21 + 3.24i)12-s + (4.22 + 2.43i)13-s + (2.40 − 2.86i)14-s + (−3.57 + 3.48i)15-s + (−3.90 + 0.866i)16-s + (3.54 − 2.04i)17-s + ⋯
L(s)  = 1  + (−0.744 − 0.667i)2-s + (0.969 − 0.247i)3-s + (0.108 + 0.994i)4-s + (−1.11 + 0.644i)5-s + (−0.886 − 0.462i)6-s + (0.0320 + 0.999i)7-s + (0.582 − 0.812i)8-s + (0.877 − 0.478i)9-s + (1.26 + 0.265i)10-s + (0.600 + 0.346i)11-s + (0.351 + 0.936i)12-s + (1.17 + 0.676i)13-s + (0.643 − 0.765i)14-s + (−0.923 + 0.900i)15-s + (−0.976 + 0.216i)16-s + (0.859 − 0.496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.986 - 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06046 + 0.0874423i\)
\(L(\frac12)\) \(\approx\) \(1.06046 + 0.0874423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.05 + 0.943i)T \)
3 \( 1 + (-1.67 + 0.427i)T \)
7 \( 1 + (-0.0846 - 2.64i)T \)
good5 \( 1 + (2.49 - 1.44i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.99 - 1.15i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.22 - 2.43i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.54 + 2.04i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.308 - 0.534i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.29 - 4.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.811 + 1.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.64T + 31T^{2} \)
37 \( 1 + (-4.02 + 6.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.216 + 0.124i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.51 - 3.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.95T + 47T^{2} \)
53 \( 1 + (4.57 + 7.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.52T + 59T^{2} \)
61 \( 1 + 9.68iT - 61T^{2} \)
67 \( 1 + 8.17iT - 67T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (4.66 - 2.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.37iT - 79T^{2} \)
83 \( 1 + (3.53 + 6.11i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-12.0 - 6.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.3 + 5.99i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.80003135987316643566801586070, −11.39287561779827662099285870155, −9.930943708752563609604662066982, −9.163019276136771272243076976431, −8.210389406202198130765952252970, −7.58851784675873342644257933557, −6.44086561876125576775268246038, −4.02037886525872466943639920293, −3.28988503603441754478328143785, −1.86374375818963251398593496056, 1.13064925983460195778910826296, 3.62137374652770589992553083178, 4.51294540296767172766250427375, 6.21693231058246564532213568124, 7.54199234345091482561141391948, 8.190965586759323247838158001599, 8.709273151509714774603489495705, 10.01471188859762030302021843011, 10.68698165995953381963776176259, 11.90214874615746645555610673039

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