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Properties

Label 2-252-252.103-c1-0-27
Degree $2$
Conductor $252$
Sign $0.921 - 0.388i$
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  + (−0.615 + 1.27i)2-s + (1.73 + 0.0580i)3-s + (−1.24 − 1.56i)4-s + (1.66 − 0.958i)5-s + (−1.13 + 2.16i)6-s + (−0.708 − 2.54i)7-s + (2.75 − 0.619i)8-s + (2.99 + 0.200i)9-s + (0.199 + 2.70i)10-s + (−0.178 − 0.103i)11-s + (−2.06 − 2.78i)12-s + (−0.960 − 0.554i)13-s + (3.68 + 0.665i)14-s + (2.93 − 1.56i)15-s + (−0.909 + 3.89i)16-s + (4.54 − 2.62i)17-s + ⋯
L(s)  = 1  + (−0.435 + 0.900i)2-s + (0.999 + 0.0335i)3-s + (−0.621 − 0.783i)4-s + (0.742 − 0.428i)5-s + (−0.464 + 0.885i)6-s + (−0.267 − 0.963i)7-s + (0.975 − 0.218i)8-s + (0.997 + 0.0669i)9-s + (0.0630 + 0.855i)10-s + (−0.0538 − 0.0311i)11-s + (−0.594 − 0.803i)12-s + (−0.266 − 0.153i)13-s + (0.984 + 0.177i)14-s + (0.756 − 0.403i)15-s + (−0.227 + 0.973i)16-s + (1.10 − 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.921 - 0.388i$
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: \((2,\ 252,\ (\ :1/2),\ 0.921 - 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40615 + 0.284414i\)
\(L(\frac12)\) \(\approx\) \(1.40615 + 0.284414i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.615 - 1.27i)T \)
3 \( 1 + (-1.73 - 0.0580i)T \)
7 \( 1 + (0.708 + 2.54i)T \)
good5 \( 1 + (-1.66 + 0.958i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.178 + 0.103i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.960 + 0.554i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.54 + 2.62i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.49 - 4.32i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.39 - 2.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.97 - 3.41i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.08T + 31T^{2} \)
37 \( 1 + (2.74 - 4.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.57 - 4.37i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.498 - 0.288i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.99T + 47T^{2} \)
53 \( 1 + (3.76 + 6.51i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 10.2iT - 61T^{2} \)
67 \( 1 + 8.46iT - 67T^{2} \)
71 \( 1 - 3.72iT - 71T^{2} \)
73 \( 1 + (3.45 - 1.99i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 17.4iT - 79T^{2} \)
83 \( 1 + (-6.42 - 11.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.60 + 1.50i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.35 - 4.24i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.51592597852517169302872178015, −10.57509201309785722922799042884, −9.825699892740220059471750845243, −9.279849126881197649953529122447, −8.028161441186143423044746352806, −7.45045678860613223327703631598, −6.21335352500566004812889332512, −4.97083236826773834459208420459, −3.63989045707973662081659576233, −1.52532839180983222435144942022, 2.04825453601859994534967932626, 2.81853574809902322475863564909, 4.19728623422829585146794307271, 5.88733963270184651885236028626, 7.34821069622515700626004376619, 8.440321941246821694424324896071, 9.219251910563159683275258612329, 9.958225470915676566435348389579, 10.76126715542789715279308494600, 12.21300504968367568991399224311

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