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Properties

Label 2-252-63.59-c1-0-1
Degree $2$
Conductor $252$
Sign $0.381 - 0.924i$
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  + (−1.60 − 0.649i)3-s − 2.96·5-s + (2.38 + 1.14i)7-s + (2.15 + 2.08i)9-s + 4.72i·11-s + (3.54 + 2.04i)13-s + (4.76 + 1.92i)15-s + (0.835 − 1.44i)17-s + (−4.25 + 2.45i)19-s + (−3.08 − 3.38i)21-s + 4.91i·23-s + 3.82·25-s + (−2.11 − 4.74i)27-s + (0.238 − 0.137i)29-s + (−1.38 + 0.801i)31-s + ⋯
L(s)  = 1  + (−0.927 − 0.374i)3-s − 1.32·5-s + (0.901 + 0.433i)7-s + (0.719 + 0.694i)9-s + 1.42i·11-s + (0.981 + 0.566i)13-s + (1.23 + 0.497i)15-s + (0.202 − 0.350i)17-s + (−0.975 + 0.563i)19-s + (−0.673 − 0.739i)21-s + 1.02i·23-s + 0.764·25-s + (−0.406 − 0.913i)27-s + (0.0442 − 0.0255i)29-s + (−0.249 + 0.143i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.381 - 0.924i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.381 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593801 + 0.397355i\)
\(L(\frac12)\) \(\approx\) \(0.593801 + 0.397355i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 + 0.649i)T \)
7 \( 1 + (-2.38 - 1.14i)T \)
good5 \( 1 + 2.96T + 5T^{2} \)
11 \( 1 - 4.72iT - 11T^{2} \)
13 \( 1 + (-3.54 - 2.04i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.835 + 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.25 - 2.45i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.91iT - 23T^{2} \)
29 \( 1 + (-0.238 + 0.137i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.38 - 0.801i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.55 + 6.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.22 - 9.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.49 - 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.707 + 0.408i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.37 - 2.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.23 + 3.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.80 + 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (-13.6 - 7.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.15 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.03 - 6.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.60 + 7.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.00 - 4.04i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.16218711593994932866873869923, −11.33744173462191866188202491852, −10.77805280763496718586779141906, −9.310147486998931259361856740409, −7.964273703664534441805470895565, −7.43800938459405721340937693223, −6.21136074769236182264447630851, −4.87159499693056306293360807039, −4.05353445818109786354387418646, −1.72115399198710913501927001541, 0.68510219330231687961062531812, 3.59185170081974633967349128685, 4.41999664402837367329464049164, 5.66794034663862783932071507822, 6.79860217323318566439958785802, 8.099927914216843641194906961146, 8.653224586983970219303154348645, 10.49344358060274901104420082383, 11.01005795928004657753088320333, 11.55515576196800596752027556142

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