Properties

Label 2-6035-1.1-c1-0-303
Degree $2$
Conductor $6035$
Sign $-1$
Analytic cond. $48.1897$
Root an. cond. $6.94188$
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  + 2.19·2-s − 2.37·3-s + 2.82·4-s + 5-s − 5.21·6-s − 1.56·7-s + 1.82·8-s + 2.63·9-s + 2.19·10-s − 2.91·11-s − 6.71·12-s + 3.27·13-s − 3.44·14-s − 2.37·15-s − 1.65·16-s − 17-s + 5.78·18-s + 0.0652·19-s + 2.82·20-s + 3.72·21-s − 6.39·22-s + 8.58·23-s − 4.32·24-s + 25-s + 7.19·26-s + 0.872·27-s − 4.43·28-s + ⋯
L(s)  = 1  + 1.55·2-s − 1.37·3-s + 1.41·4-s + 0.447·5-s − 2.12·6-s − 0.592·7-s + 0.643·8-s + 0.877·9-s + 0.694·10-s − 0.877·11-s − 1.93·12-s + 0.908·13-s − 0.921·14-s − 0.612·15-s − 0.414·16-s − 0.242·17-s + 1.36·18-s + 0.0149·19-s + 0.632·20-s + 0.812·21-s − 1.36·22-s + 1.79·23-s − 0.882·24-s + 0.200·25-s + 1.41·26-s + 0.167·27-s − 0.838·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6035\)    =    \(5 \cdot 17 \cdot 71\)
Sign: $-1$
Analytic conductor: \(48.1897\)
Root analytic conductor: \(6.94188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6035,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
17 \( 1 + T \)
71 \( 1 - T \)
good2 \( 1 - 2.19T + 2T^{2} \)
3 \( 1 + 2.37T + 3T^{2} \)
7 \( 1 + 1.56T + 7T^{2} \)
11 \( 1 + 2.91T + 11T^{2} \)
13 \( 1 - 3.27T + 13T^{2} \)
19 \( 1 - 0.0652T + 19T^{2} \)
23 \( 1 - 8.58T + 23T^{2} \)
29 \( 1 + 5.29T + 29T^{2} \)
31 \( 1 - 6.46T + 31T^{2} \)
37 \( 1 + 3.09T + 37T^{2} \)
41 \( 1 - 8.02T + 41T^{2} \)
43 \( 1 + 13.0T + 43T^{2} \)
47 \( 1 + 0.641T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 - 8.92T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + 1.40T + 67T^{2} \)
73 \( 1 - 3.08T + 73T^{2} \)
79 \( 1 - 2.11T + 79T^{2} \)
83 \( 1 + 6.22T + 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + 7.29T + 97T^{2} \)
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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

−7.18703895293235777515566628226, −6.51886347186819296507992182836, −6.20107323437305219337270918405, −5.38666672982905551913765450449, −5.08400088331957324176616253235, −4.32234909870375438196084070017, −3.29770399523568415446778195630, −2.72742374847866004327714002743, −1.39903618295165023001117896380, 0, 1.39903618295165023001117896380, 2.72742374847866004327714002743, 3.29770399523568415446778195630, 4.32234909870375438196084070017, 5.08400088331957324176616253235, 5.38666672982905551913765450449, 6.20107323437305219337270918405, 6.51886347186819296507992182836, 7.18703895293235777515566628226

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