Properties

Label 2-6035-1.1-c1-0-263
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  + 0.487·2-s + 0.235·3-s − 1.76·4-s + 5-s + 0.115·6-s − 1.09·7-s − 1.83·8-s − 2.94·9-s + 0.487·10-s − 1.06·11-s − 0.415·12-s + 3.57·13-s − 0.535·14-s + 0.235·15-s + 2.62·16-s − 17-s − 1.43·18-s − 0.200·19-s − 1.76·20-s − 0.258·21-s − 0.519·22-s + 2.04·23-s − 0.433·24-s + 25-s + 1.74·26-s − 1.40·27-s + 1.93·28-s + ⋯
L(s)  = 1  + 0.345·2-s + 0.136·3-s − 0.880·4-s + 0.447·5-s + 0.0470·6-s − 0.414·7-s − 0.649·8-s − 0.981·9-s + 0.154·10-s − 0.320·11-s − 0.119·12-s + 0.990·13-s − 0.142·14-s + 0.0609·15-s + 0.656·16-s − 0.242·17-s − 0.338·18-s − 0.0460·19-s − 0.393·20-s − 0.0564·21-s − 0.110·22-s + 0.426·23-s − 0.0884·24-s + 0.200·25-s + 0.341·26-s − 0.269·27-s + 0.365·28-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 \( 1 - 0.487T + 2T^{2} \)
3 \( 1 - 0.235T + 3T^{2} \)
7 \( 1 + 1.09T + 7T^{2} \)
11 \( 1 + 1.06T + 11T^{2} \)
13 \( 1 - 3.57T + 13T^{2} \)
19 \( 1 + 0.200T + 19T^{2} \)
23 \( 1 - 2.04T + 23T^{2} \)
29 \( 1 - 1.57T + 29T^{2} \)
31 \( 1 - 8.58T + 31T^{2} \)
37 \( 1 + 0.400T + 37T^{2} \)
41 \( 1 + 4.45T + 41T^{2} \)
43 \( 1 - 1.47T + 43T^{2} \)
47 \( 1 + 5.56T + 47T^{2} \)
53 \( 1 + 0.707T + 53T^{2} \)
59 \( 1 - 4.46T + 59T^{2} \)
61 \( 1 + 4.33T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
73 \( 1 + 2.93T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 5.73T + 89T^{2} \)
97 \( 1 - 2.53T + 97T^{2} \)
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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

−8.042089860983930503543111714612, −6.75217327421421267887622706126, −6.20704024187498287314432027116, −5.52891788614439656713044585174, −4.90446637522889569709231748042, −4.03449343615039788762235734603, −3.20308509126918151880261521698, −2.63849388389185261618826224654, −1.22826114953618817127554434832, 0, 1.22826114953618817127554434832, 2.63849388389185261618826224654, 3.20308509126918151880261521698, 4.03449343615039788762235734603, 4.90446637522889569709231748042, 5.52891788614439656713044585174, 6.20704024187498287314432027116, 6.75217327421421267887622706126, 8.042089860983930503543111714612

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