Properties

Label 2-6034-1.1-c1-0-138
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 − 2.80·3-s + 4-s + 0.235·5-s − 2.80·6-s − 7-s + 8-s + 4.84·9-s + 0.235·10-s + 0.907·11-s − 2.80·12-s + 1.15·13-s − 14-s − 0.660·15-s + 16-s − 0.425·17-s + 4.84·18-s − 3.51·19-s + 0.235·20-s + 2.80·21-s + 0.907·22-s + 3.92·23-s − 2.80·24-s − 4.94·25-s + 1.15·26-s − 5.16·27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.105·5-s − 1.14·6-s − 0.377·7-s + 0.353·8-s + 1.61·9-s + 0.0745·10-s + 0.273·11-s − 0.808·12-s + 0.321·13-s − 0.267·14-s − 0.170·15-s + 0.250·16-s − 0.103·17-s + 1.14·18-s − 0.807·19-s + 0.0527·20-s + 0.611·21-s + 0.193·22-s + 0.818·23-s − 0.571·24-s − 0.988·25-s + 0.227·26-s − 0.993·27-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 \( 1 + 2.80T + 3T^{2} \)
5 \( 1 - 0.235T + 5T^{2} \)
11 \( 1 - 0.907T + 11T^{2} \)
13 \( 1 - 1.15T + 13T^{2} \)
17 \( 1 + 0.425T + 17T^{2} \)
19 \( 1 + 3.51T + 19T^{2} \)
23 \( 1 - 3.92T + 23T^{2} \)
29 \( 1 + 4.86T + 29T^{2} \)
31 \( 1 + 2.03T + 31T^{2} \)
37 \( 1 - 4.43T + 37T^{2} \)
41 \( 1 + 2.98T + 41T^{2} \)
43 \( 1 - 9.80T + 43T^{2} \)
47 \( 1 + 9.95T + 47T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 0.501T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 - 9.16T + 71T^{2} \)
73 \( 1 + 3.75T + 73T^{2} \)
79 \( 1 - 1.44T + 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 - 0.670T + 89T^{2} \)
97 \( 1 - 7.45T + 97T^{2} \)
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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

−7.37446021237606282140662560961, −6.65290176048710449952487727829, −6.22626496584578982646491009020, −5.60478925130686490074619815151, −4.97912053220391451980858660867, −4.20280509558618213322187028957, −3.52488780872170085212077455492, −2.27964863830811446150941135765, −1.22605592256699952433387709721, 0, 1.22605592256699952433387709721, 2.27964863830811446150941135765, 3.52488780872170085212077455492, 4.20280509558618213322187028957, 4.97912053220391451980858660867, 5.60478925130686490074619815151, 6.22626496584578982646491009020, 6.65290176048710449952487727829, 7.37446021237606282140662560961

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