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Properties

Label 2-444-111.14-c1-0-11
Degree $2$
Conductor $444$
Sign $-0.952 - 0.305i$
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  + (−0.526 − 1.64i)3-s + (−0.425 + 1.58i)5-s + (−1.24 − 2.15i)7-s + (−2.44 + 1.73i)9-s − 5.53·11-s + (1.40 + 0.375i)13-s + (2.84 − 0.134i)15-s + (−3.55 + 0.951i)17-s + (−1.73 − 0.463i)19-s + (−2.89 + 3.18i)21-s + (−1.79 − 1.79i)23-s + (1.99 + 1.15i)25-s + (4.15 + 3.11i)27-s + (0.643 − 0.643i)29-s + (−5.86 − 5.86i)31-s + ⋯
L(s)  = 1  + (−0.304 − 0.952i)3-s + (−0.190 + 0.709i)5-s + (−0.469 − 0.812i)7-s + (−0.814 + 0.579i)9-s − 1.66·11-s + (0.388 + 0.104i)13-s + (0.733 − 0.0347i)15-s + (−0.861 + 0.230i)17-s + (−0.397 − 0.106i)19-s + (−0.631 + 0.694i)21-s + (−0.373 − 0.373i)23-s + (0.398 + 0.230i)25-s + (0.799 + 0.600i)27-s + (0.119 − 0.119i)29-s + (−1.05 − 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $-0.952 - 0.305i$
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: \((2,\ 444,\ (\ :1/2),\ -0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0257386 + 0.164595i\)
\(L(\frac12)\) \(\approx\) \(0.0257386 + 0.164595i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.526 + 1.64i)T \)
37 \( 1 + (4.93 - 3.55i)T \)
good5 \( 1 + (0.425 - 1.58i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.24 + 2.15i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 5.53T + 11T^{2} \)
13 \( 1 + (-1.40 - 0.375i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.55 - 0.951i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.73 + 0.463i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.79 + 1.79i)T + 23iT^{2} \)
29 \( 1 + (-0.643 + 0.643i)T - 29iT^{2} \)
31 \( 1 + (5.86 + 5.86i)T + 31iT^{2} \)
41 \( 1 + (1.33 + 2.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.92 - 6.92i)T - 43iT^{2} \)
47 \( 1 - 7.23iT - 47T^{2} \)
53 \( 1 + (3.61 + 2.08i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.30 + 2.49i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.753 + 2.81i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-9.23 + 5.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.7 + 6.78i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 + (10.2 + 2.73i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (5.13 + 2.96i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.69 + 6.33i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.50 - 6.50i)T - 97iT^{2} \)
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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

−10.85878075833429517560038312836, −9.993125324529847504081457352741, −8.503517480197817374857935153560, −7.69423600469857103339523437033, −6.90233200830832338702130204728, −6.16377278527710837398002094923, −4.90212448909831804274838981086, −3.37259598281023038615766390370, −2.18543110236300763763582524515, −0.099422970585650369998707811719, 2.55091237703202325855394520353, 3.81873492929221013412364570971, 5.12828602781019601456190719034, 5.51720923172170578454746781364, 6.86706201297923721898523119785, 8.453834403413993315963695535941, 8.751374273506030750740318564642, 9.903599319526519105529049115870, 10.62066011539824903105306153013, 11.49202523793516405754335595765

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