Properties

Label 2-2175-435.434-c0-0-14
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

\(L(\frac{1}{2})\) \(\approx\) \(1.037497836\)
\(L(\frac12)\) \(\approx\) \(1.037497836\)
\(L(1)\) 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

−8.985718094428736906211688887319, −7.78349697091661654264294015595, −7.29777507738232775289147129373, −6.71099184684778262035520630373, −5.71941715779001714511477013836, −4.57936027810556299834866804461, −3.59654560385860847177916778641, −2.63449234108729369617435247005, −1.87380803251759499117765622318, −0.67921307165896043946117356753, 2.30281055037571072244007421497, 3.02107189708964287810078851249, 4.25660107422862552127551683387, 5.35820709991137623248855350025, 5.62122507145720993748543290258, 6.30057081842881805967489518156, 7.61857097647929372284978671225, 8.062396109058887228114842168033, 8.797058562281212487504187566425, 9.487931363368921099215234554045

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