Properties

Label 2-945-1.1-c1-0-4
Degree $2$
Conductor $945$
Sign $1$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s − 1.82·4-s + 5-s − 7-s + 1.58·8-s − 0.414·10-s − 2.41·11-s − 0.414·13-s + 0.414·14-s + 3·16-s − 0.828·17-s + 4.41·19-s − 1.82·20-s + 0.999·22-s + 4.82·23-s + 25-s + 0.171·26-s + 1.82·28-s − 4·29-s + 6·31-s − 4.41·32-s + 0.343·34-s − 35-s + 8.48·37-s − 1.82·38-s + 1.58·40-s − 2.17·41-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.914·4-s + 0.447·5-s − 0.377·7-s + 0.560·8-s − 0.130·10-s − 0.727·11-s − 0.114·13-s + 0.110·14-s + 0.750·16-s − 0.200·17-s + 1.01·19-s − 0.408·20-s + 0.213·22-s + 1.00·23-s + 0.200·25-s + 0.0336·26-s + 0.345·28-s − 0.742·29-s + 1.07·31-s − 0.780·32-s + 0.0588·34-s − 0.169·35-s + 1.39·37-s − 0.296·38-s + 0.250·40-s − 0.339·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.069464015\)
\(L(\frac12)\) \(\approx\) \(1.069464015\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
good2 \( 1 + 0.414T + 2T^{2} \)
11 \( 1 + 2.41T + 11T^{2} \)
13 \( 1 + 0.414T + 13T^{2} \)
17 \( 1 + 0.828T + 17T^{2} \)
19 \( 1 - 4.41T + 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 - 4.65T + 43T^{2} \)
47 \( 1 + 9.48T + 47T^{2} \)
53 \( 1 - 8.89T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 - 2.17T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 9.17T + 79T^{2} \)
83 \( 1 - 9.48T + 83T^{2} \)
89 \( 1 - 2.65T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−9.791268141694709254995948243740, −9.418679887553456665556912496201, −8.440810768690363647788760765134, −7.68924573585406078381363536238, −6.69514942262324026107291338691, −5.52166826004903773727527928290, −4.91871515518784025867642200364, −3.72785437125309116310886621011, −2.56236852206653238634458799461, −0.873412705238580479693108341594, 0.873412705238580479693108341594, 2.56236852206653238634458799461, 3.72785437125309116310886621011, 4.91871515518784025867642200364, 5.52166826004903773727527928290, 6.69514942262324026107291338691, 7.68924573585406078381363536238, 8.440810768690363647788760765134, 9.418679887553456665556912496201, 9.791268141694709254995948243740

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