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Properties

Label 2-2175-435.434-c0-0-10
Degree $2$
Conductor $2175$
Sign $-0.894 + 0.447i$
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  − i·2-s − i·3-s − 6-s + i·7-s − i·8-s − 9-s + 11-s − i·13-s + 14-s − 16-s − i·17-s + i·18-s + 21-s − i·22-s − 24-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 6-s + i·7-s − i·8-s − 9-s + 11-s − i·13-s + 14-s − 16-s − i·17-s + i·18-s + 21-s − i·22-s − 24-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2175\)    =    \(3 \cdot 5^{2} \cdot 29\)
Sign: $-0.894 + 0.447i$
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: \((2,\ 2175,\ (\ :0),\ -0.894 + 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.299551278\)
\(L(\frac12)\) \(\approx\) \(1.299551278\)
\(L(1)\) 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 \( 1 + iT - T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 2T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T + T^{2} \)
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

−8.907894277955556905030710725497, −8.332023881021735066899812635991, −7.25003020581366123991314221918, −6.65687536901623601143291557610, −5.86540235829576364993373479037, −4.96294781581387227309729325178, −3.52547111520363988912200447408, −2.81430228398164299149514404071, −2.03124391828388729288183065624, −0.947433202032195862022651083637, 1.73563806907604143759868119808, 3.22630213458761895300968613762, 4.17436563751000314271022318241, 4.70306895354367200128885015168, 5.80345566680843606687673426517, 6.53638663714794155429031410279, 7.04188134689925459582475490368, 8.120545201009249455083349146010, 8.664226170490811445719073850017, 9.472444300959686171008626663092

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