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Properties

Label 2-444-37.25-c1-0-3
Degree $2$
Conductor $444$
Sign $0.997 + 0.0746i$
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  + (−0.766 − 0.642i)3-s + (0.322 + 0.885i)5-s + (2.90 − 1.05i)7-s + (0.173 + 0.984i)9-s + (2.09 + 3.63i)11-s + (−5.13 − 0.904i)13-s + (0.322 − 0.885i)15-s + (4.79 − 0.846i)17-s + (1.59 − 1.90i)19-s + (−2.90 − 1.05i)21-s + (2.93 + 1.69i)23-s + (3.14 − 2.64i)25-s + (0.500 − 0.866i)27-s + (8.90 − 5.13i)29-s + 10.3i·31-s + ⋯
L(s)  = 1  + (−0.442 − 0.371i)3-s + (0.144 + 0.396i)5-s + (1.09 − 0.399i)7-s + (0.0578 + 0.328i)9-s + (0.632 + 1.09i)11-s + (−1.42 − 0.250i)13-s + (0.0832 − 0.228i)15-s + (1.16 − 0.205i)17-s + (0.366 − 0.436i)19-s + (−0.633 − 0.230i)21-s + (0.612 + 0.353i)23-s + (0.629 − 0.528i)25-s + (0.0962 − 0.166i)27-s + (1.65 − 0.954i)29-s + 1.86i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.997 + 0.0746i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ 0.997 + 0.0746i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41440 - 0.0528740i\)
\(L(\frac12)\) \(\approx\) \(1.41440 - 0.0528740i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (-0.637 + 6.04i)T \)
good5 \( 1 + (-0.322 - 0.885i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-2.90 + 1.05i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-2.09 - 3.63i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.13 + 0.904i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-4.79 + 0.846i)T + (15.9 - 5.81i)T^{2} \)
19 \( 1 + (-1.59 + 1.90i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (-2.93 - 1.69i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.90 + 5.13i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
41 \( 1 + (0.710 - 4.03i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 3.16iT - 43T^{2} \)
47 \( 1 + (1.35 - 2.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.3 + 3.76i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.33 + 3.66i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (6.40 + 1.12i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (13.3 - 4.85i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-1.43 - 1.20i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 7.61T + 73T^{2} \)
79 \( 1 + (3.61 + 9.93i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.87 - 16.3i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (4.29 - 11.7i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (10.7 + 6.18i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.11076694382653201923546543850, −10.26282228637861443587282173507, −9.516141569421078486349891757697, −8.104221663291988750184385985140, −7.32064578710888603130512890544, −6.66884327003666601537422854172, −5.15230698067122315182579913793, −4.60620266501851452307214342388, −2.79341088142357302090460009371, −1.34184768584753162023517311369, 1.26708490390930846081223604588, 3.05893287236297171177410970043, 4.58704084474301907596951981427, 5.25031895750618909652785402904, 6.23342018474891220108149869296, 7.55253021487556273223969602130, 8.473597523263288569607415801994, 9.330053901299663505195334001971, 10.24769752182546552792325084344, 11.24232953200776951822862215531

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