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Properties

Label 2-252-252.223-c1-0-28
Degree $2$
Conductor $252$
Sign $0.893 - 0.449i$
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  + (0.648 + 1.25i)2-s + (0.758 − 1.55i)3-s + (−1.15 + 1.63i)4-s + (2.66 − 1.53i)5-s + (2.44 − 0.0569i)6-s + (−1.15 + 2.37i)7-s + (−2.80 − 0.397i)8-s + (−1.84 − 2.36i)9-s + (3.66 + 2.35i)10-s + (3.64 + 2.10i)11-s + (1.66 + 3.04i)12-s + (2.97 − 1.71i)13-s + (−3.74 + 0.0882i)14-s + (−0.374 − 5.31i)15-s + (−1.31 − 3.77i)16-s − 2.29i·17-s + ⋯
L(s)  = 1  + (0.458 + 0.888i)2-s + (0.437 − 0.898i)3-s + (−0.579 + 0.815i)4-s + (1.19 − 0.688i)5-s + (0.999 − 0.0232i)6-s + (−0.437 + 0.899i)7-s + (−0.990 − 0.140i)8-s + (−0.616 − 0.787i)9-s + (1.15 + 0.743i)10-s + (1.10 + 0.635i)11-s + (0.479 + 0.877i)12-s + (0.824 − 0.476i)13-s + (−0.999 + 0.0235i)14-s + (−0.0966 − 1.37i)15-s + (−0.329 − 0.944i)16-s − 0.555i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.893 - 0.449i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.893 - 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84616 + 0.438243i\)
\(L(\frac12)\) \(\approx\) \(1.84616 + 0.438243i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.648 - 1.25i)T \)
3 \( 1 + (-0.758 + 1.55i)T \)
7 \( 1 + (1.15 - 2.37i)T \)
good5 \( 1 + (-2.66 + 1.53i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.64 - 2.10i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.97 + 1.71i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.29iT - 17T^{2} \)
19 \( 1 + 1.70T + 19T^{2} \)
23 \( 1 + (4.68 - 2.70i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.76 - 8.25i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.22 + 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.04T + 37T^{2} \)
41 \( 1 + (4.43 - 2.56i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.89 + 1.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.20 - 3.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + (4.44 + 7.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.57 + 2.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.53 + 5.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.48iT - 71T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 + (-8.87 - 5.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.0572 + 0.0991i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.61iT - 89T^{2} \)
97 \( 1 + (5.53 + 3.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.56809688669030604218070466324, −11.69070390796766813006276278847, −9.527837006823966624673443937611, −9.125114938864295972746750946990, −8.215777748192700762184073217785, −6.89625231185019151093569829415, −6.08002239452007305543102691298, −5.34190762288268794363847566926, −3.54712987329578390388916212762, −1.90010377776520608241449102353, 1.96303816776974551110984041429, 3.47357742453442903152759429527, 4.14315963758290783220345028097, 5.80257594493572426008394241817, 6.52698425039179279687428018925, 8.586472331560172249904355567387, 9.449947867764486616125190724050, 10.26013023168988774230661714361, 10.73964127577862680802279355469, 11.72934199269475874560076810720

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