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Properties

Label 2-252-28.19-c1-0-17
Degree $2$
Conductor $252$
Sign $-0.916 - 0.399i$
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  + (0.272 − 1.38i)2-s + (−1.85 − 0.755i)4-s + (−2.12 − 1.22i)5-s + (−2.63 + 0.272i)7-s + (−1.55 + 2.36i)8-s + (−2.27 + 2.61i)10-s + (−1.09 + 0.632i)11-s − 2.99i·13-s + (−0.337 + 3.72i)14-s + (2.85 + 2.79i)16-s + (−1.58 + 0.916i)17-s + (2.07 − 3.60i)19-s + (3.00 + 3.87i)20-s + (0.579 + 1.69i)22-s + (−5.83 − 3.36i)23-s + ⋯
L(s)  = 1  + (0.192 − 0.981i)2-s + (−0.925 − 0.377i)4-s + (−0.949 − 0.548i)5-s + (−0.994 + 0.102i)7-s + (−0.548 + 0.836i)8-s + (−0.720 + 0.826i)10-s + (−0.330 + 0.190i)11-s − 0.831i·13-s + (−0.0903 + 0.995i)14-s + (0.714 + 0.699i)16-s + (−0.385 + 0.222i)17-s + (0.477 − 0.826i)19-s + (0.672 + 0.866i)20-s + (0.123 + 0.360i)22-s + (−1.21 − 0.702i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.916 - 0.399i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.916 - 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101066 + 0.485164i\)
\(L(\frac12)\) \(\approx\) \(0.101066 + 0.485164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.272 + 1.38i)T \)
3 \( 1 \)
7 \( 1 + (2.63 - 0.272i)T \)
good5 \( 1 + (2.12 + 1.22i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.09 - 0.632i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.99iT - 13T^{2} \)
17 \( 1 + (1.58 - 0.916i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.07 + 3.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.83 + 3.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 + (4.71 + 8.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.75 - 6.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.08iT - 41T^{2} \)
43 \( 1 - 6.27iT - 43T^{2} \)
47 \( 1 + (-3.67 + 6.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0358 + 0.0620i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.68 - 2.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.61 + 5.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.43 + 1.40i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + (-7.01 + 4.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.54 - 0.891i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.33T + 83T^{2} \)
89 \( 1 + (7.42 + 4.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.10iT - 97T^{2} \)
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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

−11.72076070559852928424724091714, −10.56587853171830758620526164862, −9.794592486501520656437337906476, −8.712535651478265908503582152224, −7.85724056616243720316139160257, −6.25783644937171220415123436576, −4.91768762226585871039423192228, −3.87650682769617184217023651112, −2.68428477137963704997372154108, −0.36219522303963726007297558965, 3.27633804344056507388961554465, 4.15709167695639846963672137177, 5.66402505474332467439114834058, 6.76973829465164327394961577188, 7.43014105906402848764767731902, 8.502431564270471196645562160477, 9.539985890528176805946359143660, 10.56954974110546945412998324922, 11.92147985163243007518655746195, 12.54291407936011278431211659870

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