Properties

Label 2-6034-1.1-c1-0-60
Degree $2$
Conductor $6034$
Sign $-1$
Analytic cond. $48.1817$
Root an. cond. $6.94130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.804·3-s + 4-s − 3.18·5-s + 0.804·6-s − 7-s − 8-s − 2.35·9-s + 3.18·10-s − 6.09·11-s − 0.804·12-s + 4.76·13-s + 14-s + 2.55·15-s + 16-s + 4.44·17-s + 2.35·18-s − 3.99·19-s − 3.18·20-s + 0.804·21-s + 6.09·22-s − 5.05·23-s + 0.804·24-s + 5.13·25-s − 4.76·26-s + 4.30·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.464·3-s + 0.5·4-s − 1.42·5-s + 0.328·6-s − 0.377·7-s − 0.353·8-s − 0.784·9-s + 1.00·10-s − 1.83·11-s − 0.232·12-s + 1.32·13-s + 0.267·14-s + 0.660·15-s + 0.250·16-s + 1.07·17-s + 0.554·18-s − 0.915·19-s − 0.711·20-s + 0.175·21-s + 1.29·22-s − 1.05·23-s + 0.164·24-s + 1.02·25-s − 0.935·26-s + 0.828·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6034\)    =    \(2 \cdot 7 \cdot 431\)
Sign: $-1$
Analytic conductor: \(48.1817\)
Root analytic conductor: \(6.94130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6034,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
431 \( 1 + T \)
good3 \( 1 + 0.804T + 3T^{2} \)
5 \( 1 + 3.18T + 5T^{2} \)
11 \( 1 + 6.09T + 11T^{2} \)
13 \( 1 - 4.76T + 13T^{2} \)
17 \( 1 - 4.44T + 17T^{2} \)
19 \( 1 + 3.99T + 19T^{2} \)
23 \( 1 + 5.05T + 23T^{2} \)
29 \( 1 - 1.73T + 29T^{2} \)
31 \( 1 - 2.54T + 31T^{2} \)
37 \( 1 - 0.204T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 5.54T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 - 0.201T + 53T^{2} \)
59 \( 1 - 7.13T + 59T^{2} \)
61 \( 1 - 7.83T + 61T^{2} \)
67 \( 1 + 2.92T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 4.06T + 79T^{2} \)
83 \( 1 + 7.35T + 83T^{2} \)
89 \( 1 - 1.32T + 89T^{2} \)
97 \( 1 - 9.61T + 97T^{2} \)
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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

−7.972640713185883852261686272546, −7.26408870785679960043961403920, −6.20798620342098498674768014978, −5.82710075653279335602809226229, −4.88371414536842366380713004183, −3.90045770041498420918377945885, −3.18817003168099956963712098937, −2.39447295996792283886839873248, −0.809788001044415387845941703292, 0, 0.809788001044415387845941703292, 2.39447295996792283886839873248, 3.18817003168099956963712098937, 3.90045770041498420918377945885, 4.88371414536842366380713004183, 5.82710075653279335602809226229, 6.20798620342098498674768014978, 7.26408870785679960043961403920, 7.972640713185883852261686272546

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