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Properties

Label 2-444-12.11-c1-0-2
Degree $2$
Conductor $444$
Sign $-0.933 - 0.357i$
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  + (−1.03 + 0.964i)2-s + (0.504 − 1.65i)3-s + (0.139 − 1.99i)4-s + 4.11i·5-s + (1.07 + 2.20i)6-s + 2.35i·7-s + (1.77 + 2.19i)8-s + (−2.49 − 1.67i)9-s + (−3.96 − 4.25i)10-s − 6.51·11-s + (−3.23 − 1.23i)12-s − 0.0134·13-s + (−2.27 − 2.43i)14-s + (6.81 + 2.07i)15-s + (−3.96 − 0.558i)16-s − 4.36i·17-s + ⋯
L(s)  = 1  + (−0.731 + 0.681i)2-s + (0.291 − 0.956i)3-s + (0.0699 − 0.997i)4-s + 1.83i·5-s + (0.439 + 0.898i)6-s + 0.890i·7-s + (0.629 + 0.777i)8-s + (−0.830 − 0.557i)9-s + (−1.25 − 1.34i)10-s − 1.96·11-s + (−0.933 − 0.357i)12-s − 0.00372·13-s + (−0.607 − 0.651i)14-s + (1.76 + 0.535i)15-s + (−0.990 − 0.139i)16-s − 1.05i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $-0.933 - 0.357i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ -0.933 - 0.357i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0923448 + 0.499569i\)
\(L(\frac12)\) \(\approx\) \(0.0923448 + 0.499569i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 - 0.964i)T \)
3 \( 1 + (-0.504 + 1.65i)T \)
37 \( 1 - T \)
good5 \( 1 - 4.11iT - 5T^{2} \)
7 \( 1 - 2.35iT - 7T^{2} \)
11 \( 1 + 6.51T + 11T^{2} \)
13 \( 1 + 0.0134T + 13T^{2} \)
17 \( 1 + 4.36iT - 17T^{2} \)
19 \( 1 - 1.88iT - 19T^{2} \)
23 \( 1 + 3.70T + 23T^{2} \)
29 \( 1 - 6.11iT - 29T^{2} \)
31 \( 1 - 0.859iT - 31T^{2} \)
41 \( 1 - 1.33iT - 41T^{2} \)
43 \( 1 - 3.41iT - 43T^{2} \)
47 \( 1 - 5.35T + 47T^{2} \)
53 \( 1 - 9.92iT - 53T^{2} \)
59 \( 1 - 9.22T + 59T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 + 2.48iT - 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 5.93T + 73T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 - 4.54T + 83T^{2} \)
89 \( 1 + 4.03iT - 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−11.25139036369740730274948444482, −10.55356670673164807076136535724, −9.689303127550035329947535383517, −8.497622084872395977615758312507, −7.62472283022496608435638885414, −7.16691628328287297454631656279, −6.12812262115094709154921688656, −5.42292679477485706662672088930, −2.89728297880592851490607000362, −2.29224121767927246558098984529, 0.36019255872311586572816449662, 2.21913314818386888498847005296, 3.84147164095032347582823344573, 4.57792393983579051438832569355, 5.57735447248161634749341040084, 7.72516024875237680586861712745, 8.204537422720575271483499722775, 8.945034668664199165159994413379, 10.00492321926042845183188269918, 10.35271767304797813674283637172

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