Properties

Label 2-6026-1.1-c1-0-9
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 − 0.471·3-s + 4-s + 1.82·5-s + 0.471·6-s − 1.64·7-s − 8-s − 2.77·9-s − 1.82·10-s − 6.17·11-s − 0.471·12-s − 3.42·13-s + 1.64·14-s − 0.859·15-s + 16-s − 5.99·17-s + 2.77·18-s − 2.80·19-s + 1.82·20-s + 0.776·21-s + 6.17·22-s + 23-s + 0.471·24-s − 1.68·25-s + 3.42·26-s + 2.72·27-s − 1.64·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.272·3-s + 0.5·4-s + 0.814·5-s + 0.192·6-s − 0.621·7-s − 0.353·8-s − 0.925·9-s − 0.575·10-s − 1.86·11-s − 0.136·12-s − 0.950·13-s + 0.439·14-s − 0.221·15-s + 0.250·16-s − 1.45·17-s + 0.654·18-s − 0.644·19-s + 0.407·20-s + 0.169·21-s + 1.31·22-s + 0.208·23-s + 0.0963·24-s − 0.337·25-s + 0.671·26-s + 0.524·27-s − 0.310·28-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

\(L(1)\) \(\approx\) \(0.2268210475\)
\(L(\frac12)\) \(\approx\) \(0.2268210475\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.471T + 3T^{2} \)
5 \( 1 - 1.82T + 5T^{2} \)
7 \( 1 + 1.64T + 7T^{2} \)
11 \( 1 + 6.17T + 11T^{2} \)
13 \( 1 + 3.42T + 13T^{2} \)
17 \( 1 + 5.99T + 17T^{2} \)
19 \( 1 + 2.80T + 19T^{2} \)
29 \( 1 + 3.31T + 29T^{2} \)
31 \( 1 - 3.32T + 31T^{2} \)
37 \( 1 + 8.75T + 37T^{2} \)
41 \( 1 - 5.09T + 41T^{2} \)
43 \( 1 + 6.77T + 43T^{2} \)
47 \( 1 + 4.23T + 47T^{2} \)
53 \( 1 - 4.22T + 53T^{2} \)
59 \( 1 - 5.10T + 59T^{2} \)
61 \( 1 - 6.07T + 61T^{2} \)
67 \( 1 - 8.73T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 5.69T + 73T^{2} \)
79 \( 1 - 0.0898T + 79T^{2} \)
83 \( 1 + 1.67T + 83T^{2} \)
89 \( 1 + 7.08T + 89T^{2} \)
97 \( 1 + 6.48T + 97T^{2} \)
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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

−8.286435679320853885659781610584, −7.33702822235909200301343359435, −6.70203342159280199110558466379, −5.99410555803691164158750920430, −5.35019173302441457559114509584, −4.72673716781258490103006251804, −3.31723088642282412218216783875, −2.43184726327102436928798451665, −2.11791722287105439648882061637, −0.25447927845808531973715817117, 0.25447927845808531973715817117, 2.11791722287105439648882061637, 2.43184726327102436928798451665, 3.31723088642282412218216783875, 4.72673716781258490103006251804, 5.35019173302441457559114509584, 5.99410555803691164158750920430, 6.70203342159280199110558466379, 7.33702822235909200301343359435, 8.286435679320853885659781610584

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