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Properties

Label 2-252-252.139-c1-0-38
Degree $2$
Conductor $252$
Sign $-0.809 + 0.587i$
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  + (0.139 − 1.40i)2-s + (1.60 − 0.649i)3-s + (−1.96 − 0.392i)4-s + (−1.35 − 0.782i)5-s + (−0.690 − 2.35i)6-s + (−2.11 − 1.58i)7-s + (−0.825 + 2.70i)8-s + (2.15 − 2.08i)9-s + (−1.28 + 1.79i)10-s + (2.65 − 1.53i)11-s + (−3.40 + 0.644i)12-s + (−4.17 − 2.41i)13-s + (−2.52 + 2.75i)14-s + (−2.68 − 0.375i)15-s + (3.69 + 1.53i)16-s + 0.603i·17-s + ⋯
L(s)  = 1  + (0.0985 − 0.995i)2-s + (0.926 − 0.375i)3-s + (−0.980 − 0.196i)4-s + (−0.605 − 0.349i)5-s + (−0.282 − 0.959i)6-s + (−0.800 − 0.599i)7-s + (−0.291 + 0.956i)8-s + (0.718 − 0.695i)9-s + (−0.407 + 0.568i)10-s + (0.800 − 0.462i)11-s + (−0.982 + 0.186i)12-s + (−1.15 − 0.669i)13-s + (−0.675 + 0.737i)14-s + (−0.692 − 0.0968i)15-s + (0.922 + 0.384i)16-s + 0.146i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.809 + 0.587i$
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: \((2,\ 252,\ (\ :1/2),\ -0.809 + 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.407572 - 1.25639i\)
\(L(\frac12)\) \(\approx\) \(0.407572 - 1.25639i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.139 + 1.40i)T \)
3 \( 1 + (-1.60 + 0.649i)T \)
7 \( 1 + (2.11 + 1.58i)T \)
good5 \( 1 + (1.35 + 0.782i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.65 + 1.53i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.17 + 2.41i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.603iT - 17T^{2} \)
19 \( 1 - 5.64T + 19T^{2} \)
23 \( 1 + (-7.81 - 4.51i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.38 - 5.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.860 - 1.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.45T + 37T^{2} \)
41 \( 1 + (2.08 + 1.20i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.473 + 0.273i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.83 - 3.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.430T + 53T^{2} \)
59 \( 1 + (2.05 - 3.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.3 + 6.53i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.39 + 0.803i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 + 1.88iT - 73T^{2} \)
79 \( 1 + (-6.41 + 3.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.801 - 1.38i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.0iT - 89T^{2} \)
97 \( 1 + (14.8 - 8.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.97691071379378015489598733026, −10.67630795069583593429439474055, −9.632274662423828426568283961461, −9.030335091515172691264077913445, −7.88732609147868663473867613236, −6.92199167247440006711714193706, −5.08308154455428524382862321263, −3.69081674839570655559971428154, −3.02156570446005826568522961480, −1.02265219499911716536181284311, 2.89627049815434368488712107051, 4.04932006888576766237090376406, 5.15758925543702226378575350936, 6.83579202613745961112659798872, 7.30178137791228054112232292651, 8.564573862359936236287420224844, 9.403830661347030952347470875012, 9.939347790553109666062559718383, 11.70824520059757740564673053943, 12.59365347250828683619486828097

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