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Properties

Label 2-444-111.14-c1-0-8
Degree $2$
Conductor $444$
Sign $-0.346 + 0.937i$
Analytic cond. $3.54535$
Root an. cond. $1.88291$
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  + (−0.795 + 1.53i)3-s + (0.826 − 3.08i)5-s + (−0.374 − 0.649i)7-s + (−1.73 − 2.44i)9-s − 3.71·11-s + (−6.34 − 1.70i)13-s + (4.08 + 3.72i)15-s + (4.32 − 1.15i)17-s + (−6.08 − 1.62i)19-s + (1.29 − 0.0603i)21-s + (1.17 + 1.17i)23-s + (−4.49 − 2.59i)25-s + (5.14 − 0.722i)27-s + (3.95 − 3.95i)29-s + (1.62 + 1.62i)31-s + ⋯
L(s)  = 1  + (−0.459 + 0.888i)3-s + (0.369 − 1.37i)5-s + (−0.141 − 0.245i)7-s + (−0.578 − 0.815i)9-s − 1.12·11-s + (−1.76 − 0.471i)13-s + (1.05 + 0.961i)15-s + (1.04 − 0.280i)17-s + (−1.39 − 0.373i)19-s + (0.283 − 0.0131i)21-s + (0.244 + 0.244i)23-s + (−0.898 − 0.518i)25-s + (0.990 − 0.139i)27-s + (0.734 − 0.734i)29-s + (0.291 + 0.291i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $-0.346 + 0.937i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ -0.346 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.375305 - 0.538873i\)
\(L(\frac12)\) \(\approx\) \(0.375305 - 0.538873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.795 - 1.53i)T \)
37 \( 1 + (5.95 + 1.22i)T \)
good5 \( 1 + (-0.826 + 3.08i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.374 + 0.649i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.71T + 11T^{2} \)
13 \( 1 + (6.34 + 1.70i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-4.32 + 1.15i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (6.08 + 1.62i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.17 - 1.17i)T + 23iT^{2} \)
29 \( 1 + (-3.95 + 3.95i)T - 29iT^{2} \)
31 \( 1 + (-1.62 - 1.62i)T + 31iT^{2} \)
41 \( 1 + (3.37 + 5.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.14 + 5.14i)T - 43iT^{2} \)
47 \( 1 - 4.04iT - 47T^{2} \)
53 \( 1 + (-6.47 - 3.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.46 + 2.00i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.60 - 9.71i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.553 + 0.319i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.302 + 0.174i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.24iT - 73T^{2} \)
79 \( 1 + (7.39 + 1.98i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (13.5 + 7.81i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.86 - 10.6i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.779 - 0.779i)T - 97iT^{2} \)
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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

−10.39329345334104751390472166669, −10.16475098701400045840061969198, −9.139300617099600540863247814186, −8.326808083801337739936427385701, −7.16204646137368846067462283347, −5.58175080603340713865610098875, −5.14045706628336830821614741440, −4.25747923445685492660748429629, −2.62377009048301271914499578136, −0.40360934318628998363558626277, 2.18000206427105708282708257007, 2.91419716562268930050181628366, 4.89836038959211046808510622068, 5.93174106931871194134168889436, 6.80818739146079284907402106310, 7.45727665439050872298284943887, 8.416169722138540569324227192197, 10.04030735070599139414390043431, 10.39413727248092678785798833808, 11.37882212595428973124836268898

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