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Properties

Label 2-444-111.14-c1-0-7
Degree $2$
Conductor $444$
Sign $0.838 + 0.544i$
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  + (1.70 − 0.284i)3-s + (0.662 − 2.47i)5-s + (1.02 + 1.77i)7-s + (2.83 − 0.971i)9-s − 2.87·11-s + (3.09 + 0.828i)13-s + (0.429 − 4.41i)15-s + (1.46 − 0.391i)17-s + (−0.779 − 0.208i)19-s + (2.25 + 2.74i)21-s + (−5.03 − 5.03i)23-s + (−1.35 − 0.779i)25-s + (4.57 − 2.46i)27-s + (−2.13 + 2.13i)29-s + (0.584 + 0.584i)31-s + ⋯
L(s)  = 1  + (0.986 − 0.164i)3-s + (0.296 − 1.10i)5-s + (0.388 + 0.672i)7-s + (0.946 − 0.323i)9-s − 0.867·11-s + (0.857 + 0.229i)13-s + (0.110 − 1.13i)15-s + (0.354 − 0.0949i)17-s + (−0.178 − 0.0479i)19-s + (0.493 + 0.599i)21-s + (−1.04 − 1.04i)23-s + (−0.270 − 0.155i)25-s + (0.880 − 0.474i)27-s + (−0.396 + 0.396i)29-s + (0.105 + 0.105i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.838 + 0.544i$
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: \((2,\ 444,\ (\ :1/2),\ 0.838 + 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98266 - 0.587415i\)
\(L(\frac12)\) \(\approx\) \(1.98266 - 0.587415i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.284i)T \)
37 \( 1 + (2.93 + 5.32i)T \)
good5 \( 1 + (-0.662 + 2.47i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.02 - 1.77i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.87T + 11T^{2} \)
13 \( 1 + (-3.09 - 0.828i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.46 + 0.391i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.779 + 0.208i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.03 + 5.03i)T + 23iT^{2} \)
29 \( 1 + (2.13 - 2.13i)T - 29iT^{2} \)
31 \( 1 + (-0.584 - 0.584i)T + 31iT^{2} \)
41 \( 1 + (-4.03 - 6.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.93 - 4.93i)T - 43iT^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 + (0.0121 + 0.00703i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.81 - 1.02i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.418 - 1.56i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (12.3 - 7.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-13.4 + 7.77i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.50iT - 73T^{2} \)
79 \( 1 + (5.78 + 1.55i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-5.18 - 2.99i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.0713 + 0.266i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.33 + 1.33i)T - 97iT^{2} \)
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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

−10.97110532045195116902221396044, −9.887513913200889373515789578981, −9.000400402651180476011994654607, −8.422119866340412110803644474673, −7.73018255668784664473778580148, −6.27709304728633075018651108984, −5.19096217702723145589240559396, −4.17532422334953823784961632137, −2.69890794321927407509684990810, −1.48695269638680395052452986963, 1.90734474039989295814194519082, 3.15663644800414312407187145270, 4.01672828387486937849092296575, 5.49882441730676310765672293927, 6.75287182114034263883717299002, 7.66103565600010981280053011988, 8.276962556894363778294413605779, 9.520836942114770892028026120121, 10.46675380427985705400749816707, 10.71443042391417815466898733775

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