Properties

Label 2-3234-1.1-c1-0-26
Degree $2$
Conductor $3234$
Sign $1$
Analytic cond. $25.8236$
Root an. cond. $5.08169$
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  + 2-s + 3-s + 4-s − 1.79·5-s + 6-s + 8-s + 9-s − 1.79·10-s + 11-s + 12-s + 3.63·13-s − 1.79·15-s + 16-s + 6.11·17-s + 18-s + 1.84·19-s − 1.79·20-s + 22-s − 7.37·23-s + 24-s − 1.76·25-s + 3.63·26-s + 27-s − 10.4·29-s − 1.79·30-s + 7.97·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.804·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.568·10-s + 0.301·11-s + 0.288·12-s + 1.00·13-s − 0.464·15-s + 0.250·16-s + 1.48·17-s + 0.235·18-s + 0.422·19-s − 0.402·20-s + 0.213·22-s − 1.53·23-s + 0.204·24-s − 0.353·25-s + 0.713·26-s + 0.192·27-s − 1.93·29-s − 0.328·30-s + 1.43·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(25.8236\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3234,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.537780447\)
\(L(\frac12)\) \(\approx\) \(3.537780447\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 1.79T + 5T^{2} \)
13 \( 1 - 3.63T + 13T^{2} \)
17 \( 1 - 6.11T + 17T^{2} \)
19 \( 1 - 1.84T + 19T^{2} \)
23 \( 1 + 7.37T + 23T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 - 7.97T + 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 - 8.17T + 41T^{2} \)
43 \( 1 + 2.28T + 43T^{2} \)
47 \( 1 + 0.669T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 + 0.274T + 61T^{2} \)
67 \( 1 - 3.88T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 - 3.85T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 5.27T + 83T^{2} \)
89 \( 1 - 1.98T + 89T^{2} \)
97 \( 1 - 4.12T + 97T^{2} \)
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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

−8.398972059912393218162739350148, −7.78084056162295000120016772826, −7.42385408797160163828533229805, −6.15207722259159205260635651208, −5.77907318189705000055202745145, −4.54244151956040147248065469084, −3.80075734082115028205940408739, −3.40805249083392281889684762419, −2.24289057288142240105593990981, −1.04836234215526698932321262365, 1.04836234215526698932321262365, 2.24289057288142240105593990981, 3.40805249083392281889684762419, 3.80075734082115028205940408739, 4.54244151956040147248065469084, 5.77907318189705000055202745145, 6.15207722259159205260635651208, 7.42385408797160163828533229805, 7.78084056162295000120016772826, 8.398972059912393218162739350148

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