Properties

Label 2-8034-1.1-c1-0-51
Degree $2$
Conductor $8034$
Sign $1$
Analytic cond. $64.1518$
Root an. cond. $8.00948$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 1.54·5-s + 6-s − 5.16·7-s + 8-s + 9-s + 1.54·10-s − 5.89·11-s + 12-s + 13-s − 5.16·14-s + 1.54·15-s + 16-s + 7.18·17-s + 18-s − 6.16·19-s + 1.54·20-s − 5.16·21-s − 5.89·22-s + 3.21·23-s + 24-s − 2.60·25-s + 26-s + 27-s − 5.16·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.692·5-s + 0.408·6-s − 1.95·7-s + 0.353·8-s + 0.333·9-s + 0.489·10-s − 1.77·11-s + 0.288·12-s + 0.277·13-s − 1.38·14-s + 0.399·15-s + 0.250·16-s + 1.74·17-s + 0.235·18-s − 1.41·19-s + 0.346·20-s − 1.12·21-s − 1.25·22-s + 0.670·23-s + 0.204·24-s − 0.520·25-s + 0.196·26-s + 0.192·27-s − 0.976·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
Sign: $1$
Analytic conductor: \(64.1518\)
Root analytic conductor: \(8.00948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.332618125\)
\(L(\frac12)\) \(\approx\) \(3.332618125\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
13 \( 1 - T \)
103 \( 1 + T \)
good5 \( 1 - 1.54T + 5T^{2} \)
7 \( 1 + 5.16T + 7T^{2} \)
11 \( 1 + 5.89T + 11T^{2} \)
17 \( 1 - 7.18T + 17T^{2} \)
19 \( 1 + 6.16T + 19T^{2} \)
23 \( 1 - 3.21T + 23T^{2} \)
29 \( 1 - 5.32T + 29T^{2} \)
31 \( 1 - 6.16T + 31T^{2} \)
37 \( 1 - 5.89T + 37T^{2} \)
41 \( 1 + 0.460T + 41T^{2} \)
43 \( 1 + 0.464T + 43T^{2} \)
47 \( 1 + 2.61T + 47T^{2} \)
53 \( 1 + 2.66T + 53T^{2} \)
59 \( 1 - 5.23T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 + 0.908T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 0.487T + 83T^{2} \)
89 \( 1 - 1.49T + 89T^{2} \)
97 \( 1 + 12.5T + 97T^{2} \)
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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

−8.000847594812217352034610840892, −6.82214969500251380967703085214, −6.48814705222628220070907656290, −5.70336533064966026359071072990, −5.20448328899376319974125700897, −4.13038400424278276453353028220, −3.33605085397854791372185417320, −2.77192004623231761956388825461, −2.27819681231267296188458153631, −0.75405137002776566803773516498, 0.75405137002776566803773516498, 2.27819681231267296188458153631, 2.77192004623231761956388825461, 3.33605085397854791372185417320, 4.13038400424278276453353028220, 5.20448328899376319974125700897, 5.70336533064966026359071072990, 6.48814705222628220070907656290, 6.82214969500251380967703085214, 8.000847594812217352034610840892

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