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Properties

Label 2-2175-435.434-c0-0-0
Degree $2$
Conductor $2175$
Sign $0.447 - 0.894i$
Analytic cond. $1.08546$
Root an. cond. $1.04185$
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  − 1.53i·2-s + i·3-s − 1.34·4-s + 1.53·6-s − 0.347i·7-s + 0.532i·8-s − 9-s − 1.87·11-s − 1.34i·12-s + 1.53i·13-s − 0.532·14-s − 0.532·16-s + 1.87i·17-s + 1.53i·18-s + 0.347·21-s + 2.87i·22-s + ⋯
L(s)  = 1  − 1.53i·2-s + i·3-s − 1.34·4-s + 1.53·6-s − 0.347i·7-s + 0.532i·8-s − 9-s − 1.87·11-s − 1.34i·12-s + 1.53i·13-s − 0.532·14-s − 0.532·16-s + 1.87i·17-s + 1.53i·18-s + 0.347·21-s + 2.87i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2175\)    =    \(3 \cdot 5^{2} \cdot 29\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(1.08546\)
Root analytic conductor: \(1.04185\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2175} (2174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2175,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4705855295\)
\(L(\frac12)\) \(\approx\) \(0.4705855295\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
29 \( 1 + T \)
good2 \( 1 + 1.53iT - T^{2} \)
7 \( 1 + 0.347iT - T^{2} \)
11 \( 1 + 1.87T + T^{2} \)
13 \( 1 - 1.53iT - T^{2} \)
17 \( 1 - 1.87iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 0.347iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.87iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.53T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.794664496049774914349793300550, −8.842289165057299589484617289619, −8.303877059851758213174890045515, −7.15368005895162563293798798638, −5.95599638642109390814397575020, −5.04573130507909009545061965652, −4.18238621047646188466607106595, −3.68358561489941601004838613502, −2.63092421854304962046399015711, −1.79161249331612592450483384027, 0.30197695048536623300829006736, 2.38606741449672637671138149564, 3.08517586013256609681273757950, 5.02567961140049885902652765560, 5.32266525152502721051546961128, 5.96799289749568700923953124413, 6.98750338964348973994925404829, 7.61005305031405771143167606347, 7.928101495684215172776214488396, 8.676462234252508220536862889354

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