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Properties

Label 2-4400-5.4-c1-0-23
Degree $2$
Conductor $4400$
Sign $-0.894 - 0.447i$
Analytic cond. $35.1341$
Root an. cond. $5.92740$
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  + 2.79i·3-s − 0.208i·7-s − 4.79·9-s + 11-s − i·13-s − 0.791i·17-s + 6.58·19-s + 0.582·21-s + 3.79i·23-s − 4.99i·27-s − 6.79·29-s + 8.58·31-s + 2.79i·33-s + 2.58i·37-s + 2.79·39-s + ⋯
L(s)  = 1  + 1.61i·3-s − 0.0788i·7-s − 1.59·9-s + 0.301·11-s − 0.277i·13-s − 0.191i·17-s + 1.51·19-s + 0.127·21-s + 0.790i·23-s − 0.962i·27-s − 1.26·29-s + 1.54·31-s + 0.485i·33-s + 0.424i·37-s + 0.446·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(35.1341\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4400} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.660758233\)
\(L(\frac12)\) \(\approx\) \(1.660758233\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good3 \( 1 - 2.79iT - 3T^{2} \)
7 \( 1 + 0.208iT - 7T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + 0.791iT - 17T^{2} \)
19 \( 1 - 6.58T + 19T^{2} \)
23 \( 1 - 3.79iT - 23T^{2} \)
29 \( 1 + 6.79T + 29T^{2} \)
31 \( 1 - 8.58T + 31T^{2} \)
37 \( 1 - 2.58iT - 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + 1.41iT - 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 4.20T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 7.79iT - 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 - 9.95iT - 83T^{2} \)
89 \( 1 - 0.791T + 89T^{2} \)
97 \( 1 - 6.20iT - 97T^{2} \)
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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

−8.947076518578473608542777152790, −7.994762833236819241757640961384, −7.37395342545581043961001092118, −6.26812650862192244829221147442, −5.50734546682872376662024379502, −4.91397410411789914065325994907, −4.16248956289390071705329839527, −3.41001153887090481038786686783, −2.77394191210548824842656234159, −1.19396089585866482458430720296, 0.51120410345282507646669023921, 1.47798096105292030401073893233, 2.30370194219875638942630951753, 3.19380512017666953124710520274, 4.23830960898633452174474143019, 5.39339243518740575403218658570, 5.93818089134249958150696063022, 6.89415877206793976574350537666, 7.11252583285465168772017647942, 8.001691636017243026483317449072

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