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Properties

Label 2-252-84.11-c1-0-6
Degree $2$
Conductor $252$
Sign $0.930 + 0.366i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.930 + 0.366i$
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: \((2,\ 252,\ (\ :1/2),\ 0.930 + 0.366i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06493 - 0.201956i\)
\(L(\frac12)\) \(\approx\) \(1.06493 - 0.201956i\)
\(L(\frac{3}{2})\) 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

−12.14511890216403524177462833681, −10.79437309835020561826739498725, −9.837479664937097432136221826232, −9.232759603484536332650211273178, −8.455634785857629400158582303997, −7.20991345144145124053718152951, −5.84528952664315918721583455010, −4.91162111420822667783978164080, −2.63752136973291923600430059517, −1.60210501857844799639941171937, 1.53677357176533033626721128058, 3.08025905203862541273399837564, 5.41665209790184187093703634784, 6.10955269192016403354905239395, 7.27235626568470782776503249100, 8.042279092388001713777049248646, 9.530803873768194211567830474977, 10.02977679628411757181762052745, 10.80937178092225426136116340341, 11.65895551013800008602749123393

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