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Properties

Label 2-2175-5.4-c1-0-15
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  + 2.56i·2-s − i·3-s − 4.56·4-s + 2.56·6-s − 3.12i·7-s − 6.56i·8-s − 9-s − 5.56·11-s + 4.56i·12-s + 2i·13-s + 8·14-s + 7.68·16-s + 1.12i·17-s − 2.56i·18-s + 3.12·19-s + ⋯
L(s)  = 1  + 1.81i·2-s − 0.577i·3-s − 2.28·4-s + 1.04·6-s − 1.18i·7-s − 2.31i·8-s − 0.333·9-s − 1.67·11-s + 1.31i·12-s + 0.554i·13-s + 2.13·14-s + 1.92·16-s + 0.272i·17-s − 0.603i·18-s + 0.716·19-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

\(L(1)\) \(\approx\) \(0.9179690014\)
\(L(\frac12)\) \(\approx\) \(0.9179690014\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56iT - 2T^{2} \)
7 \( 1 + 3.12iT - 7T^{2} \)
11 \( 1 + 5.56T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 1.12iT - 17T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
23 \( 1 + 2.43iT - 23T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 10.6iT - 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 4.68iT - 43T^{2} \)
47 \( 1 + 4.87iT - 47T^{2} \)
53 \( 1 - 7.56iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 - 13.3iT - 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 10.6iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 1.56iT - 83T^{2} \)
89 \( 1 - 4.24T + 89T^{2} \)
97 \( 1 - 6.68iT - 97T^{2} \)
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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

−9.046853585728283829089775159895, −8.099707705731057820353778767370, −7.77844300012635102264914450806, −7.11941902189559581219967915467, −6.46517297776241480823529265099, −5.65421265092610980269431655960, −4.83892974456055030360231832003, −4.13209492771647556333732171127, −2.78158736611961523639792596892, −0.938075068546832454543000935030, 0.41124233683428177989139806321, 2.09133606814422816123499123628, 2.76312201184365734867181228665, 3.39826119013099505134550932173, 4.55792882557521425906612668622, 5.31353517840059737139040302142, 5.78055829869145500749852374058, 7.61895413525123564445502202020, 8.272043484307526779558785349190, 9.288147464462274579651194142907

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