Properties

Label 2-252-63.38-c1-0-4
Degree $2$
Conductor $252$
Sign $0.835 - 0.549i$
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  + (1.06 + 1.36i)3-s + (0.349 − 0.605i)5-s + (2.48 − 0.903i)7-s + (−0.721 + 2.91i)9-s + (0.229 − 0.132i)11-s + (1.13 − 0.657i)13-s + (1.19 − 0.169i)15-s + (−1.86 + 3.22i)17-s + (−0.382 + 0.220i)19-s + (3.88 + 2.42i)21-s + (−4.29 − 2.48i)23-s + (2.25 + 3.90i)25-s + (−4.74 + 2.12i)27-s + (−0.273 − 0.157i)29-s − 5.60i·31-s + ⋯
L(s)  = 1  + (0.616 + 0.787i)3-s + (0.156 − 0.270i)5-s + (0.939 − 0.341i)7-s + (−0.240 + 0.970i)9-s + (0.0692 − 0.0399i)11-s + (0.315 − 0.182i)13-s + (0.309 − 0.0437i)15-s + (−0.452 + 0.783i)17-s + (−0.0877 + 0.0506i)19-s + (0.848 + 0.529i)21-s + (−0.896 − 0.517i)23-s + (0.451 + 0.781i)25-s + (−0.912 + 0.408i)27-s + (−0.0507 − 0.0292i)29-s − 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55837 + 0.466186i\)
\(L(\frac12)\) \(\approx\) \(1.55837 + 0.466186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.06 - 1.36i)T \)
7 \( 1 + (-2.48 + 0.903i)T \)
good5 \( 1 + (-0.349 + 0.605i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.229 + 0.132i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.13 + 0.657i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.86 - 3.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.382 - 0.220i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.29 + 2.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.273 + 0.157i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.60iT - 31T^{2} \)
37 \( 1 + (0.351 + 0.608i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.39 + 9.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.73 + 6.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.00T + 47T^{2} \)
53 \( 1 + (-8.51 - 4.91i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 5.65iT - 61T^{2} \)
67 \( 1 + 5.94T + 67T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + (6.66 + 3.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.39T + 79T^{2} \)
83 \( 1 + (3.72 - 6.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.59 - 9.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.18 + 5.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.04616378323783537236107367294, −10.88731427023196959096956818997, −10.36366651653179959855598112454, −9.106572597541022654975283233515, −8.408519402141003658281875201397, −7.47311496352876489771931275357, −5.82868859303551555810099997790, −4.67324569066599531114353140980, −3.73346097876549378959598862155, −1.99517888335978018592533039284, 1.67431635050185702493800927166, 2.97630413011677751586744821371, 4.61062038087198224349245622903, 6.06046282445673222498538864525, 7.06418066361020674722452901398, 8.100295400807900170666649110666, 8.822404217866770994618727188775, 9.932148174936653124891732500840, 11.26920169064906248115163991688, 11.91347953095189150862883396193

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