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Properties

Label 2-252-28.3-c1-0-2
Degree $2$
Conductor $252$
Sign $-0.192 - 0.981i$
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  + (−1.26 + 0.638i)2-s + (1.18 − 1.61i)4-s + (−0.380 + 0.219i)5-s + (−2.02 + 1.70i)7-s + (−0.464 + 2.79i)8-s + (0.339 − 0.519i)10-s + (1.83 + 1.05i)11-s + 3.84i·13-s + (1.46 − 3.44i)14-s + (−1.19 − 3.81i)16-s + (4.89 + 2.82i)17-s + (1.48 + 2.57i)19-s + (−0.0963 + 0.872i)20-s + (−2.98 − 0.164i)22-s + (−4.13 + 2.38i)23-s + ⋯
L(s)  = 1  + (−0.892 + 0.451i)2-s + (0.592 − 0.805i)4-s + (−0.170 + 0.0981i)5-s + (−0.764 + 0.644i)7-s + (−0.164 + 0.986i)8-s + (0.107 − 0.164i)10-s + (0.552 + 0.318i)11-s + 1.06i·13-s + (0.391 − 0.920i)14-s + (−0.299 − 0.954i)16-s + (1.18 + 0.684i)17-s + (0.341 + 0.591i)19-s + (−0.0215 + 0.195i)20-s + (−0.637 − 0.0350i)22-s + (−0.861 + 0.497i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.192 - 0.981i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.192 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.436701 + 0.530436i\)
\(L(\frac12)\) \(\approx\) \(0.436701 + 0.530436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 - 0.638i)T \)
3 \( 1 \)
7 \( 1 + (2.02 - 1.70i)T \)
good5 \( 1 + (0.380 - 0.219i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.83 - 1.05i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.84iT - 13T^{2} \)
17 \( 1 + (-4.89 - 2.82i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.48 - 2.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.13 - 2.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.02T + 29T^{2} \)
31 \( 1 + (-3.71 + 6.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.64 - 4.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.81iT - 41T^{2} \)
43 \( 1 + 4.38iT - 43T^{2} \)
47 \( 1 + (0.844 + 1.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.35 + 9.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.05 + 7.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.35 + 3.09i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.79 - 3.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.16iT - 71T^{2} \)
73 \( 1 + (8.69 + 5.01i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.4 - 7.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.49T + 83T^{2} \)
89 \( 1 + (-9.02 + 5.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.22iT - 97T^{2} \)
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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

−11.91927357813497068405119834274, −11.47729167528412266792897540695, −9.806729218936093741532895062166, −9.691341907612040194855654919646, −8.430081590058821864785374378605, −7.46272882304874651750283007291, −6.38452710401031064066498738431, −5.57463307642275145729467616432, −3.69760100500919173162550387944, −1.86023225272346241158289434204, 0.74415156774313237273644663285, 2.89180637606477860682756153213, 3.95327762730302146425938836115, 5.86175815093532902502745147218, 7.10883827049542725225442769563, 7.892121244174600465160590120775, 9.020193841528322343928567583657, 9.956746911990255376316902203085, 10.58132328070621941517738352311, 11.73922924470203646074659188052

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