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Properties

Label 2-252-28.19-c1-0-16
Degree $2$
Conductor $252$
Sign $-0.159 + 0.987i$
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  + (1.06 − 0.929i)2-s + (0.272 − 1.98i)4-s + (−2.12 − 1.22i)5-s + (2.63 − 0.272i)7-s + (−1.55 − 2.36i)8-s + (−3.40 + 0.667i)10-s + (1.09 − 0.632i)11-s − 2.99i·13-s + (2.55 − 2.73i)14-s + (−3.85 − 1.07i)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  + (0.753 − 0.657i)2-s + (0.136 − 0.990i)4-s + (−0.949 − 0.548i)5-s + (0.994 − 0.102i)7-s + (−0.548 − 0.836i)8-s + (−1.07 + 0.210i)10-s + (0.330 − 0.190i)11-s − 0.831i·13-s + (0.681 − 0.731i)14-s + (−0.962 − 0.269i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.159 + 0.987i$
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: \((2,\ 252,\ (\ :1/2),\ -0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09775 - 1.28917i\)
\(L(\frac12)\) \(\approx\) \(1.09775 - 1.28917i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.929i)T \)
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

−11.98474618720698575054129157429, −10.99601685241655840108952328204, −10.28515696599102177189777069327, −8.775524533848370169011872888383, −8.010384012475789522728868773503, −6.60208359515166646462897868589, −5.16577828931565750942510899931, −4.42936324390873685973407811378, −3.20106541827624145945970987185, −1.26262668655151211852510921938, 2.63619337065818592779357144419, 4.17959146924510661062528950724, 4.82965937439152877020118826160, 6.46238529673685546853229254480, 7.20140304957665284760253645730, 8.178950358975451240080466051710, 9.040335875916587708892785073015, 10.87648849591453261951513528543, 11.52142568283725767680231474363, 12.18756519209824302085690371112

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