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Properties

Label 1-4563-4563.905-r0-0-0
Degree $1$
Conductor $4563$
Sign $0.261 - 0.965i$
Analytic cond. $21.1904$
Root an. cond. $21.1904$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.367 − 0.930i)2-s + (−0.730 − 0.682i)4-s + (−0.997 − 0.0670i)5-s + (−0.265 − 0.964i)7-s + (−0.903 + 0.428i)8-s + (−0.428 + 0.903i)10-s + (−0.852 + 0.523i)11-s + (−0.994 − 0.107i)14-s + (0.0670 + 0.997i)16-s + (−0.996 − 0.0804i)17-s + (−0.866 − 0.5i)19-s + (0.682 + 0.730i)20-s + (0.173 + 0.984i)22-s + (0.766 − 0.642i)23-s + (0.991 + 0.133i)25-s + ⋯
L(s)  = 1  + (0.367 − 0.930i)2-s + (−0.730 − 0.682i)4-s + (−0.997 − 0.0670i)5-s + (−0.265 − 0.964i)7-s + (−0.903 + 0.428i)8-s + (−0.428 + 0.903i)10-s + (−0.852 + 0.523i)11-s + (−0.994 − 0.107i)14-s + (0.0670 + 0.997i)16-s + (−0.996 − 0.0804i)17-s + (−0.866 − 0.5i)19-s + (0.682 + 0.730i)20-s + (0.173 + 0.984i)22-s + (0.766 − 0.642i)23-s + (0.991 + 0.133i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $0.261 - 0.965i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (905, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ 0.261 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5011661384 - 0.3833946604i\)
\(L(\frac12)\) \(\approx\) \(0.5011661384 - 0.3833946604i\)
\(L(1)\) \(\approx\) \(0.5890688898 - 0.4152095283i\)
\(L(1)\) \(\approx\) \(0.5890688898 - 0.4152095283i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.367 - 0.930i)T \)
5 \( 1 + (-0.997 - 0.0670i)T \)
7 \( 1 + (-0.265 - 0.964i)T \)
11 \( 1 + (-0.852 + 0.523i)T \)
17 \( 1 + (-0.996 - 0.0804i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.930 - 0.367i)T \)
31 \( 1 + (0.925 + 0.379i)T \)
37 \( 1 + (-0.534 + 0.845i)T \)
41 \( 1 + (-0.757 - 0.653i)T \)
43 \( 1 + (-0.611 + 0.791i)T \)
47 \( 1 + (0.682 + 0.730i)T \)
53 \( 1 + (-0.568 + 0.822i)T \)
59 \( 1 + (0.997 + 0.0670i)T \)
61 \( 1 + (-0.0938 + 0.995i)T \)
67 \( 1 + (-0.891 - 0.452i)T \)
71 \( 1 + (-0.160 - 0.987i)T \)
73 \( 1 + (-0.979 + 0.200i)T \)
79 \( 1 + (0.226 + 0.974i)T \)
83 \( 1 + (0.757 - 0.653i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.997 + 0.0670i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.37136219066988224294723925915, −17.645080127248507919721076134081, −16.803119632609091567108745455856, −16.18287826459885376030218882013, −15.556455870593426029629966077670, −15.18295989527152131792771567189, −14.6615093442192173391508525101, −13.58056118262308693482580022330, −13.02910489075635140176896661936, −12.47457055809289637756892781807, −11.68493335244574063679769590603, −11.09455022253258819872531073409, −10.12624686978996686826774562649, −9.102037537602291523888664340964, −8.52879481300120808557406860215, −8.13620071704309510864933844364, −7.20681271710657361501357252712, −6.651636095222282388403710851354, −5.77215212153485256665544231172, −5.2027307828060346234897389787, −4.41846890294083813176875882510, −3.58435062728931206193411771311, −2.99858030658556181365548140723, −2.044046124060477782636017665181, −0.32490548562365879749041369037, 0.453673467538378235889763783722, 1.47562529699740005025430086944, 2.553339885248101537170449328586, 3.12485694056501732580578970769, 4.08124889201991735139587073215, 4.53424933428436964840184579201, 5.06258963493791492961775058793, 6.3434347605000950211501303236, 6.97458203183356134470115555638, 7.79497450969798296996010072381, 8.59728003411957522689484203502, 9.22660078310012303035700119686, 10.32162227840212210906067159645, 10.577262579733254356649126778044, 11.28007178944364204182700781497, 11.95450488252834914528400398439, 12.801263026485075440076018916521, 13.16078576972099867556046868480, 13.79770813069880578275774956186, 14.79927888542716147824434708573, 15.28055510672745862038283861805, 15.86958826020217612064404902366, 16.893861180527879981737100707862, 17.44905905295124762974682208755, 18.29898394113030795740669939102

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