Properties

Label 2-6035-1.1-c1-0-294
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  − 0.174·2-s + 1.77·3-s − 1.96·4-s + 5-s − 0.309·6-s − 0.0296·7-s + 0.692·8-s + 0.135·9-s − 0.174·10-s − 1.11·11-s − 3.48·12-s − 3.59·13-s + 0.00516·14-s + 1.77·15-s + 3.81·16-s + 17-s − 0.0236·18-s − 3.20·19-s − 1.96·20-s − 0.0524·21-s + 0.193·22-s − 2.84·23-s + 1.22·24-s + 25-s + 0.626·26-s − 5.07·27-s + 0.0583·28-s + ⋯
L(s)  = 1  − 0.123·2-s + 1.02·3-s − 0.984·4-s + 0.447·5-s − 0.126·6-s − 0.0111·7-s + 0.244·8-s + 0.0451·9-s − 0.0551·10-s − 0.334·11-s − 1.00·12-s − 0.996·13-s + 0.00138·14-s + 0.457·15-s + 0.954·16-s + 0.242·17-s − 0.00557·18-s − 0.735·19-s − 0.440·20-s − 0.0114·21-s + 0.0413·22-s − 0.593·23-s + 0.250·24-s + 0.200·25-s + 0.122·26-s − 0.976·27-s + 0.0110·28-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 \( 1 + 0.174T + 2T^{2} \)
3 \( 1 - 1.77T + 3T^{2} \)
7 \( 1 + 0.0296T + 7T^{2} \)
11 \( 1 + 1.11T + 11T^{2} \)
13 \( 1 + 3.59T + 13T^{2} \)
19 \( 1 + 3.20T + 19T^{2} \)
23 \( 1 + 2.84T + 23T^{2} \)
29 \( 1 - 6.90T + 29T^{2} \)
31 \( 1 - 9.20T + 31T^{2} \)
37 \( 1 - 2.24T + 37T^{2} \)
41 \( 1 - 1.03T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 2.37T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 6.96T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
73 \( 1 + 9.54T + 73T^{2} \)
79 \( 1 + 9.95T + 79T^{2} \)
83 \( 1 + 8.79T + 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 - 0.184T + 97T^{2} \)
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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

−7.949270080544436458928773940577, −7.31438744280279851895692619613, −6.17634872891877073636620078733, −5.60353339580290252839213400499, −4.51832858168967114129537733203, −4.26584902495832514792702443201, −2.90729636963257064422333504191, −2.63769014239143810958446685914, −1.36872521107136449433113081077, 0, 1.36872521107136449433113081077, 2.63769014239143810958446685914, 2.90729636963257064422333504191, 4.26584902495832514792702443201, 4.51832858168967114129537733203, 5.60353339580290252839213400499, 6.17634872891877073636620078733, 7.31438744280279851895692619613, 7.949270080544436458928773940577

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