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Properties

Label 2-252-63.5-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: $\chi_{252} (5, \cdot )$
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

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

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