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Properties

Label 2-252-252.103-c1-0-20
Degree $2$
Conductor $252$
Sign $0.713 - 0.700i$
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  + (−0.171 + 1.40i)2-s + (−1.50 + 0.849i)3-s + (−1.94 − 0.480i)4-s + (3.50 − 2.02i)5-s + (−0.933 − 2.26i)6-s + (2.35 − 1.19i)7-s + (1.00 − 2.64i)8-s + (1.55 − 2.56i)9-s + (2.24 + 5.27i)10-s + (−0.471 − 0.271i)11-s + (3.33 − 0.923i)12-s + (1.01 + 0.583i)13-s + (1.27 + 3.51i)14-s + (−3.57 + 6.03i)15-s + (3.53 + 1.86i)16-s + (−5.05 + 2.92i)17-s + ⋯
L(s)  = 1  + (−0.121 + 0.992i)2-s + (−0.871 + 0.490i)3-s + (−0.970 − 0.240i)4-s + (1.56 − 0.905i)5-s + (−0.381 − 0.924i)6-s + (0.891 − 0.452i)7-s + (0.356 − 0.934i)8-s + (0.518 − 0.854i)9-s + (0.709 + 1.66i)10-s + (−0.142 − 0.0819i)11-s + (0.963 − 0.266i)12-s + (0.280 + 0.161i)13-s + (0.341 + 0.940i)14-s + (−0.923 + 1.55i)15-s + (0.884 + 0.466i)16-s + (−1.22 + 0.708i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.713 - 0.700i$
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: \((2,\ 252,\ (\ :1/2),\ 0.713 - 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04690 + 0.427969i\)
\(L(\frac12)\) \(\approx\) \(1.04690 + 0.427969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.171 - 1.40i)T \)
3 \( 1 + (1.50 - 0.849i)T \)
7 \( 1 + (-2.35 + 1.19i)T \)
good5 \( 1 + (-3.50 + 2.02i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.471 + 0.271i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.01 - 0.583i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.05 - 2.92i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.16 + 2.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.39 + 0.807i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.40 - 4.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.49T + 31T^{2} \)
37 \( 1 + (1.48 - 2.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.89 + 2.25i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.89 + 1.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.58T + 47T^{2} \)
53 \( 1 + (-5.84 - 10.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 2.08iT - 61T^{2} \)
67 \( 1 - 7.09iT - 67T^{2} \)
71 \( 1 + 5.17iT - 71T^{2} \)
73 \( 1 + (-5.58 + 3.22i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.02iT - 79T^{2} \)
83 \( 1 + (2.76 + 4.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.63 + 4.40i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.8 - 7.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.35913004764897102111595561087, −10.88126848900750539349348894590, −10.16603893407720806318603849938, −9.150195006517397442457038072994, −8.487260659914782019812292962107, −6.85093573620668279647762725522, −6.00842723913332831035538890679, −5.05189008185959888154309345167, −4.45207587068437680750307484569, −1.30861269807335715382074390174, 1.66923108325185233667730161338, 2.61955282373482441448749162099, 4.79262964108180373027032118199, 5.67742653266161246598133869385, 6.74121637528378074241803790252, 8.128402710635861653819355360512, 9.420876769057147562951828910007, 10.26505024440464898534125385870, 11.06283723703802129944573049488, 11.64172876238113945134065456049

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