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Properties

Label 2-252-252.139-c1-0-2
Degree $2$
Conductor $252$
Sign $-0.152 - 0.988i$
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  + (−0.801 − 1.16i)2-s + (−0.420 + 1.68i)3-s + (−0.716 + 1.86i)4-s + (2.25 + 1.30i)5-s + (2.29 − 0.856i)6-s + (−2.51 + 0.811i)7-s + (2.74 − 0.661i)8-s + (−2.64 − 1.41i)9-s + (−0.289 − 3.67i)10-s + (−3.80 + 2.19i)11-s + (−2.83 − 1.98i)12-s + (−2.58 − 1.49i)13-s + (2.96 + 2.28i)14-s + (−3.13 + 3.24i)15-s + (−2.97 − 2.67i)16-s + 6.35i·17-s + ⋯
L(s)  = 1  + (−0.566 − 0.824i)2-s + (−0.242 + 0.970i)3-s + (−0.358 + 0.933i)4-s + (1.00 + 0.582i)5-s + (0.936 − 0.349i)6-s + (−0.951 + 0.306i)7-s + (0.972 − 0.233i)8-s + (−0.882 − 0.470i)9-s + (−0.0916 − 1.16i)10-s + (−1.14 + 0.662i)11-s + (−0.818 − 0.573i)12-s + (−0.717 − 0.414i)13-s + (0.791 + 0.610i)14-s + (−0.810 + 0.837i)15-s + (−0.743 − 0.668i)16-s + 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.152 - 0.988i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.152 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430920 + 0.502655i\)
\(L(\frac12)\) \(\approx\) \(0.430920 + 0.502655i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.801 + 1.16i)T \)
3 \( 1 + (0.420 - 1.68i)T \)
7 \( 1 + (2.51 - 0.811i)T \)
good5 \( 1 + (-2.25 - 1.30i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.80 - 2.19i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.58 + 1.49i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.35iT - 17T^{2} \)
19 \( 1 - 4.43T + 19T^{2} \)
23 \( 1 + (-1.50 - 0.868i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.886 - 1.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.41 - 2.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + (-6.88 - 3.97i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.939 - 0.542i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.58 - 7.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.31T + 53T^{2} \)
59 \( 1 + (-4.10 + 7.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.57 - 2.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.92 + 2.26i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 - 7.69iT - 73T^{2} \)
79 \( 1 + (-11.7 + 6.77i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.518 - 0.897i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.167iT - 89T^{2} \)
97 \( 1 + (5.35 - 3.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.29661057694391669468277237882, −10.91442609352137422571210018638, −10.19620994334763236470255792784, −9.854707406447511613422709053497, −8.926056959813027197946793504304, −7.55122520702011306569028578105, −6.11575666525030931855099930713, −4.99779293122182632713624638742, −3.39275600702924140782284080067, −2.44768114994552859199640847309, 0.62347300910520445230606232571, 2.50299049099712985312986251045, 5.18610896918082474255400115562, 5.73042149712917547013960916720, 6.93773620740593656223481938465, 7.57943799843459681749366898215, 8.902363132531270326594070194264, 9.613093220770841662428044225970, 10.56476985353940820086933176019, 11.87386443677498744500295361038

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