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Properties

Label 2-252-252.103-c1-0-22
Degree $2$
Conductor $252$
Sign $0.999 - 0.0274i$
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  + (−1.33 − 0.477i)2-s + (1.43 + 0.964i)3-s + (1.54 + 1.27i)4-s + (3.01 − 1.74i)5-s + (−1.45 − 1.97i)6-s + (1.90 + 1.83i)7-s + (−1.44 − 2.43i)8-s + (1.13 + 2.77i)9-s + (−4.84 + 0.876i)10-s + (−1.71 − 0.987i)11-s + (0.992 + 3.31i)12-s + (−5.52 − 3.18i)13-s + (−1.65 − 3.35i)14-s + (6.01 + 0.404i)15-s + (0.762 + 3.92i)16-s + (1.04 − 0.600i)17-s + ⋯
L(s)  = 1  + (−0.941 − 0.337i)2-s + (0.830 + 0.556i)3-s + (0.771 + 0.636i)4-s + (1.34 − 0.778i)5-s + (−0.593 − 0.804i)6-s + (0.719 + 0.694i)7-s + (−0.511 − 0.859i)8-s + (0.379 + 0.925i)9-s + (−1.53 + 0.277i)10-s + (−0.515 − 0.297i)11-s + (0.286 + 0.958i)12-s + (−1.53 − 0.884i)13-s + (−0.442 − 0.896i)14-s + (1.55 + 0.104i)15-s + (0.190 + 0.981i)16-s + (0.252 − 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.999 - 0.0274i$
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: \((2,\ 252,\ (\ :1/2),\ 0.999 - 0.0274i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31688 + 0.0180706i\)
\(L(\frac12)\) \(\approx\) \(1.31688 + 0.0180706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.477i)T \)
3 \( 1 + (-1.43 - 0.964i)T \)
7 \( 1 + (-1.90 - 1.83i)T \)
good5 \( 1 + (-3.01 + 1.74i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.71 + 0.987i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.52 + 3.18i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.04 + 0.600i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.25 - 3.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.511 + 0.295i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.00 + 3.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.27T + 31T^{2} \)
37 \( 1 + (-0.506 + 0.877i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.54 + 2.62i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.66 - 1.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.349T + 47T^{2} \)
53 \( 1 + (-3.01 - 5.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 14.5T + 59T^{2} \)
61 \( 1 - 3.95iT - 61T^{2} \)
67 \( 1 + 4.89iT - 67T^{2} \)
71 \( 1 - 1.47iT - 71T^{2} \)
73 \( 1 + (-9.61 + 5.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 0.642iT - 79T^{2} \)
83 \( 1 + (1.00 + 1.74i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.27 + 1.88i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.8 + 8.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.08619206545483210521597592009, −10.59975944712260578952401652379, −9.954802446787332584689211548046, −9.268424304104155556233452623917, −8.363063735737669424599292126926, −7.69739831265508363173519410093, −5.82488545130559772376479990030, −4.82124260552796574707259638458, −2.79606284266091844894493897681, −1.89267319697099975746434271531, 1.78441912036518622288918846086, 2.63812765105664554100278475077, 4.99612467956002824533074617178, 6.58525678923910623391996788232, 7.09330334907293801441006857869, 8.043580078123871342921454137722, 9.254566402116682738939533370500, 9.921514879797990987398603262914, 10.68904292923454461873221971300, 11.89963113467699876462268689729

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