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Properties

Label 2-252-84.23-c1-0-13
Degree $2$
Conductor $252$
Sign $-0.502 + 0.864i$
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  + (0.171 + 1.40i)2-s + (−1.94 + 0.480i)4-s + (−3.35 − 1.93i)5-s + (−1.03 + 2.43i)7-s + (−1.00 − 2.64i)8-s + (2.14 − 5.03i)10-s + (−1.73 − 3.00i)11-s − 0.296·13-s + (−3.59 − 1.03i)14-s + (3.53 − 1.86i)16-s + (−1.35 + 0.783i)17-s + (−6.12 − 3.53i)19-s + (7.43 + 2.14i)20-s + (3.92 − 2.95i)22-s + (−2.71 + 4.70i)23-s + ⋯
L(s)  = 1  + (0.121 + 0.992i)2-s + (−0.970 + 0.240i)4-s + (−1.49 − 0.865i)5-s + (−0.390 + 0.920i)7-s + (−0.356 − 0.934i)8-s + (0.677 − 1.59i)10-s + (−0.523 − 0.905i)11-s − 0.0822·13-s + (−0.961 − 0.275i)14-s + (0.884 − 0.466i)16-s + (−0.329 + 0.190i)17-s + (−1.40 − 0.811i)19-s + (1.66 + 0.479i)20-s + (0.835 − 0.628i)22-s + (−0.566 + 0.981i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.502 + 0.864i$
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: \((2,\ 252,\ (\ :1/2),\ -0.502 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00679390 - 0.0118070i\)
\(L(\frac12)\) \(\approx\) \(0.00679390 - 0.0118070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.171 - 1.40i)T \)
3 \( 1 \)
7 \( 1 + (1.03 - 2.43i)T \)
good5 \( 1 + (3.35 + 1.93i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.73 + 3.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.296T + 13T^{2} \)
17 \( 1 + (1.35 - 0.783i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.12 + 3.53i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.71 - 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.85iT - 29T^{2} \)
31 \( 1 + (-2.43 + 1.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.25 - 2.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.55iT - 41T^{2} \)
43 \( 1 - 0.682iT - 43T^{2} \)
47 \( 1 + (1.18 - 2.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.540 + 0.311i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.42 + 7.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.33 - 2.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.19 + 5.30i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.539T + 71T^{2} \)
73 \( 1 + (3.69 + 6.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.33 + 3.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.15T + 83T^{2} \)
89 \( 1 + (-10.1 - 5.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.84T + 97T^{2} \)
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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

−12.00702293068547951083132147379, −10.92197023557192562381440884167, −9.274153519372452498054007135638, −8.537146059736685181038560229801, −7.990754103996369532684584110140, −6.73727993229154138534350806203, −5.53816170628382833764084134712, −4.54529426334549852399129947107, −3.36822383015791042321556037946, −0.01007324111414765899141014524, 2.55291430765516996817464568716, 3.91638619948890235878306184849, 4.44637603811287929083663651700, 6.47234982661415930501789647737, 7.59840497374081659498781462484, 8.403627946591875521503878244886, 10.03156291731725707660233139265, 10.49345569591951301040019258344, 11.36700216626816943108834787545, 12.26406123879070189027679219027

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