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Properties

Label 2-252-84.11-c1-0-4
Degree $2$
Conductor $252$
Sign $0.0349 - 0.999i$
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  + (1.30 + 0.553i)2-s + (1.38 + 1.44i)4-s + (−3.35 + 1.93i)5-s + (1.03 + 2.43i)7-s + (1.00 + 2.64i)8-s + (−5.43 + 0.663i)10-s + (1.73 − 3.00i)11-s − 0.296·13-s + (−0.00435 + 3.74i)14-s + (−0.151 + 3.99i)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  + (0.920 + 0.391i)2-s + (0.693 + 0.720i)4-s + (−1.49 + 0.865i)5-s + (0.390 + 0.920i)7-s + (0.356 + 0.934i)8-s + (−1.71 + 0.209i)10-s + (0.523 − 0.905i)11-s − 0.0822·13-s + (−0.00116 + 0.999i)14-s + (−0.0379 + 0.999i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0349 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0349 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.0349 - 0.999i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.0349 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27331 + 1.22951i\)
\(L(\frac12)\) \(\approx\) \(1.27331 + 1.22951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 - 0.553i)T \)
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

−11.97328394820299703490262038523, −11.53579298925607183679626157031, −11.04861944322054264585571901004, −9.135729324665011134881903259554, −7.999793997554318020700444323807, −7.32044644079330583013284569458, −6.21681119278377397239928051447, −5.01508470289398796402995745418, −3.72099549558854085032779057728, −2.83302139464307207040874117363, 1.24587168567365712544895238667, 3.52817365779600940256935681225, 4.35623886012299860774326503387, 5.13805067820405424776629009810, 6.97087878961859234340569140305, 7.58791287070643293845038822804, 8.888261261432534205089357933794, 10.22672210015043215275564901430, 11.14539430322143556557454627227, 12.05704239834253381504231320399

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