Properties

Label 2-2175-5.4-c1-0-16
Degree $2$
Conductor $2175$
Sign $-0.894 - 0.447i$
Analytic cond. $17.3674$
Root an. cond. $4.16742$
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  + i·2-s i·3-s + 4-s + 6-s + 4i·7-s + 3i·8-s − 9-s − 4·11-s i·12-s − 6i·13-s − 4·14-s − 16-s + 6i·17-s i·18-s + 4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s + 0.5·4-s + 0.408·6-s + 1.51i·7-s + 1.06i·8-s − 0.333·9-s − 1.20·11-s − 0.288i·12-s − 1.66i·13-s − 1.06·14-s − 0.250·16-s + 1.45i·17-s − 0.235i·18-s + 0.917·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2175\)    =    \(3 \cdot 5^{2} \cdot 29\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(17.3674\)
Root analytic conductor: \(4.16742\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2175} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2175,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.318344501\)
\(L(\frac12)\) \(\approx\) \(1.318344501\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
29 \( 1 + T \)
good2 \( 1 - iT - 2T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−9.057313080397504636498094233117, −8.309521536089128087636987815792, −7.83926668980204009205845404972, −7.21947970859282073331256958176, −6.06387341895018438278700109296, −5.52295313792686514588730913045, −5.31534244219314263306376428164, −3.31328980230958298925207899687, −2.61956463322222460130096929243, −1.68432752889768692649237709781, 0.41368165828808100915162555769, 1.79621527603148595073834414844, 2.87508803831621090236192534290, 3.70715960809490838707105568047, 4.49913024472449427293520856697, 5.30132582455757463426531845302, 6.65187256612330474512955645408, 7.15767418905591412524372299714, 7.78878616378449729901467018796, 9.070889185073528998810106383266

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