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Properties

Label 2-252-252.31-c1-0-20
Degree $2$
Conductor $252$
Sign $0.993 - 0.110i$
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  + (1.36 − 0.366i)2-s − 1.73·3-s + (1.73 − i)4-s + 3.46i·5-s + (−2.36 + 0.633i)6-s + (1.73 − 2i)7-s + (1.99 − 2i)8-s + 2.99·9-s + (1.26 + 4.73i)10-s + 2i·11-s + (−2.99 + 1.73i)12-s + (4.5 + 2.59i)13-s + (1.63 − 3.36i)14-s − 5.99i·15-s + (1.99 − 3.46i)16-s + (−4.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s − 1.00·3-s + (0.866 − 0.5i)4-s + 1.54i·5-s + (−0.965 + 0.258i)6-s + (0.654 − 0.755i)7-s + (0.707 − 0.707i)8-s + 0.999·9-s + (0.400 + 1.49i)10-s + 0.603i·11-s + (−0.866 + 0.499i)12-s + (1.24 + 0.720i)13-s + (0.436 − 0.899i)14-s − 1.54i·15-s + (0.499 − 0.866i)16-s + (−1.09 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.993 - 0.110i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78347 + 0.0992256i\)
\(L(\frac12)\) \(\approx\) \(1.78347 + 0.0992256i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 + 1.73T \)
7 \( 1 + (-1.73 + 2i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.866 + 1.5i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.59 + 4.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.52 - 5.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.866 - 1.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.79 + 4.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 + (10.5 + 6.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.59 - 1.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.59 + 4.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.5 + 4.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.56683276150457130872344004295, −11.32346222591890417612047557831, −10.69037288934932126710156537480, −9.723133459401484134241219616730, −7.46964435319284695911721295976, −6.82887323972644808475642856110, −6.06009229468338389542901405614, −4.64204744619510715749249436556, −3.72957285996390457444583341717, −1.93478447149307208096553483780, 1.57616325475670918537306844807, 3.92342805970372108328068343570, 5.04324386480462238592226540436, 5.58876086320798678931715335107, 6.56952448695260347795331798014, 8.268518419870145807628433761855, 8.708761099332485275511802288944, 10.61014969538009744171508661488, 11.32798507645177187864312787009, 12.23618690142603735237266744532

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