Properties

Label 2-6026-1.1-c1-0-45
Degree $2$
Conductor $6026$
Sign $1$
Analytic cond. $48.1178$
Root an. cond. $6.93670$
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 − 2.71·3-s + 4-s − 2.38·5-s + 2.71·6-s − 4.13·7-s − 8-s + 4.34·9-s + 2.38·10-s + 5.88·11-s − 2.71·12-s − 3.00·13-s + 4.13·14-s + 6.46·15-s + 16-s + 6.38·17-s − 4.34·18-s + 4.95·19-s − 2.38·20-s + 11.2·21-s − 5.88·22-s + 23-s + 2.71·24-s + 0.691·25-s + 3.00·26-s − 3.65·27-s − 4.13·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.56·3-s + 0.5·4-s − 1.06·5-s + 1.10·6-s − 1.56·7-s − 0.353·8-s + 1.44·9-s + 0.754·10-s + 1.77·11-s − 0.782·12-s − 0.833·13-s + 1.10·14-s + 1.66·15-s + 0.250·16-s + 1.54·17-s − 1.02·18-s + 1.13·19-s − 0.533·20-s + 2.44·21-s − 1.25·22-s + 0.208·23-s + 0.553·24-s + 0.138·25-s + 0.589·26-s − 0.703·27-s − 0.782·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6026\)    =    \(2 \cdot 23 \cdot 131\)
Sign: $1$
Analytic conductor: \(48.1178\)
Root analytic conductor: \(6.93670\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6026,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5248822194\)
\(L(\frac12)\) \(\approx\) \(0.5248822194\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
23 \( 1 - T \)
131 \( 1 + T \)
good3 \( 1 + 2.71T + 3T^{2} \)
5 \( 1 + 2.38T + 5T^{2} \)
7 \( 1 + 4.13T + 7T^{2} \)
11 \( 1 - 5.88T + 11T^{2} \)
13 \( 1 + 3.00T + 13T^{2} \)
17 \( 1 - 6.38T + 17T^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
29 \( 1 - 9.90T + 29T^{2} \)
31 \( 1 + 5.84T + 31T^{2} \)
37 \( 1 + 5.08T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 4.90T + 43T^{2} \)
47 \( 1 - 7.91T + 47T^{2} \)
53 \( 1 + 6.91T + 53T^{2} \)
59 \( 1 - 2.15T + 59T^{2} \)
61 \( 1 - 1.81T + 61T^{2} \)
67 \( 1 + 3.88T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 - 6.84T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + 3.32T + 83T^{2} \)
89 \( 1 + 6.85T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−7.86093872572732452106358430274, −7.08416310289336908811922749204, −6.88533921538062245788099113267, −6.00140874967013510626753815384, −5.53002112781936393727378794438, −4.39521653951707869551193479432, −3.66673035763273032341702385361, −2.93391240548889786077551453549, −1.14971110015921201042484740747, −0.56450619043106361714706016307, 0.56450619043106361714706016307, 1.14971110015921201042484740747, 2.93391240548889786077551453549, 3.66673035763273032341702385361, 4.39521653951707869551193479432, 5.53002112781936393727378794438, 6.00140874967013510626753815384, 6.88533921538062245788099113267, 7.08416310289336908811922749204, 7.86093872572732452106358430274

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