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Properties

Label 2-252-84.23-c1-0-0
Degree $2$
Conductor $252$
Sign $-0.617 - 0.786i$
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  + (−1.41 − 0.0556i)2-s + (1.99 + 0.157i)4-s + (−1.80 − 1.04i)5-s + (−1.89 + 1.84i)7-s + (−2.80 − 0.333i)8-s + (2.48 + 1.57i)10-s + (2.13 + 3.69i)11-s − 4.80·13-s + (2.77 − 2.50i)14-s + (3.95 + 0.627i)16-s + (−2.77 + 1.60i)17-s + (2.43 + 1.40i)19-s + (−3.42 − 2.35i)20-s + (−2.80 − 5.34i)22-s + (−2.33 + 4.03i)23-s + ⋯
L(s)  = 1  + (−0.999 − 0.0393i)2-s + (0.996 + 0.0786i)4-s + (−0.805 − 0.465i)5-s + (−0.715 + 0.698i)7-s + (−0.993 − 0.117i)8-s + (0.787 + 0.496i)10-s + (0.643 + 1.11i)11-s − 1.33·13-s + (0.742 − 0.669i)14-s + (0.987 + 0.156i)16-s + (−0.673 + 0.388i)17-s + (0.559 + 0.323i)19-s + (−0.766 − 0.527i)20-s + (−0.599 − 1.13i)22-s + (−0.485 + 0.841i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.617 - 0.786i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.136069 + 0.279792i\)
\(L(\frac12)\) \(\approx\) \(0.136069 + 0.279792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0556i)T \)
3 \( 1 \)
7 \( 1 + (1.89 - 1.84i)T \)
good5 \( 1 + (1.80 + 1.04i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.13 - 3.69i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.80T + 13T^{2} \)
17 \( 1 + (2.77 - 1.60i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.43 - 1.40i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.33 - 4.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.87iT - 29T^{2} \)
31 \( 1 + (8.90 - 5.14i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.136 - 0.237i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.387iT - 41T^{2} \)
43 \( 1 - 0.907iT - 43T^{2} \)
47 \( 1 + (-3.92 + 6.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.1 + 5.85i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.85 + 3.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.01 + 6.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.21 + 0.701i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (-6.14 - 10.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.715 + 0.413i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.69T + 83T^{2} \)
89 \( 1 + (2.61 + 1.51i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−12.20532612086879002751492276205, −11.58130203375667136118123200167, −10.18408681578854816700861171348, −9.448601451290308676432169540243, −8.672856962338980742699898395646, −7.49786805896205496817009896704, −6.79605139569058559479881227376, −5.31079137558811212620966466533, −3.69855548632041184956926436394, −2.06480003200494188193656637871, 0.31712553162300814197771474132, 2.74337693577496743114507544227, 3.98144078929634212571889354758, 5.96194465961815162552905129053, 7.09518589253178697751764128697, 7.59234152077637526419302193875, 8.905854275989714889763396394698, 9.717831850514598528788019216796, 10.75042493351829083850076735161, 11.45950126674556370185082332453

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