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Properties

Label 2-252-252.103-c1-0-2
Degree $2$
Conductor $252$
Sign $-0.819 + 0.573i$
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  + (0.667 + 1.24i)2-s + (−1.72 + 0.113i)3-s + (−1.10 + 1.66i)4-s + (−0.627 + 0.362i)5-s + (−1.29 − 2.07i)6-s + (−2.64 − 0.0721i)7-s + (−2.81 − 0.273i)8-s + (2.97 − 0.393i)9-s + (−0.870 − 0.540i)10-s + (−0.501 − 0.289i)11-s + (1.72 − 3.00i)12-s + (−3.35 − 1.93i)13-s + (−1.67 − 3.34i)14-s + (1.04 − 0.697i)15-s + (−1.53 − 3.69i)16-s + (−3.20 + 1.84i)17-s + ⋯
L(s)  = 1  + (0.471 + 0.881i)2-s + (−0.997 + 0.0657i)3-s + (−0.554 + 0.831i)4-s + (−0.280 + 0.162i)5-s + (−0.528 − 0.848i)6-s + (−0.999 − 0.0272i)7-s + (−0.995 − 0.0968i)8-s + (0.991 − 0.131i)9-s + (−0.275 − 0.171i)10-s + (−0.151 − 0.0873i)11-s + (0.499 − 0.866i)12-s + (−0.930 − 0.537i)13-s + (−0.447 − 0.894i)14-s + (0.269 − 0.180i)15-s + (−0.384 − 0.923i)16-s + (−0.776 + 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.819 + 0.573i$
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: \((2,\ 252,\ (\ :1/2),\ -0.819 + 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105427 - 0.334324i\)
\(L(\frac12)\) \(\approx\) \(0.105427 - 0.334324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.667 - 1.24i)T \)
3 \( 1 + (1.72 - 0.113i)T \)
7 \( 1 + (2.64 + 0.0721i)T \)
good5 \( 1 + (0.627 - 0.362i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.501 + 0.289i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.35 + 1.93i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.20 - 1.84i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.04 - 3.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.47 + 2.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.56 - 7.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.848T + 31T^{2} \)
37 \( 1 + (1.73 - 3.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.854 + 0.493i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.9 - 6.32i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 + (2.70 + 4.69i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.387T + 59T^{2} \)
61 \( 1 + 0.891iT - 61T^{2} \)
67 \( 1 + 3.15iT - 67T^{2} \)
71 \( 1 - 7.56iT - 71T^{2} \)
73 \( 1 + (10.1 - 5.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 3.39iT - 79T^{2} \)
83 \( 1 + (3.46 + 6.00i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.37 + 1.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.32 + 5.38i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.71502297469887911608417248383, −11.96814279347184279707488225729, −10.71806863728906598485765599377, −9.803879708014388558818283186823, −8.577065096434560562442455120225, −7.24409222153973673294437635674, −6.61270315690204511658057033307, −5.59246894045719359928864069044, −4.55886705177543961576226121736, −3.25915123478032511859426026165, 0.25622694901127546857628664830, 2.46042612844425737870874857783, 4.13805995577404164608466967354, 5.00948650709956426009776721940, 6.24395198435099334989901222035, 7.13287606516216383172137958549, 8.993307032022237426331771604205, 9.853151226195134482069545126032, 10.66744404366115493696349699379, 11.71363258995330569832707789180

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