Properties

Label 2-6026-1.1-c1-0-124
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 − 1.90·3-s + 4-s + 3.90·5-s + 1.90·6-s + 1.35·7-s − 8-s + 0.630·9-s − 3.90·10-s + 4.12·11-s − 1.90·12-s + 0.627·13-s − 1.35·14-s − 7.44·15-s + 16-s + 5.89·17-s − 0.630·18-s + 6.12·19-s + 3.90·20-s − 2.57·21-s − 4.12·22-s + 23-s + 1.90·24-s + 10.2·25-s − 0.627·26-s + 4.51·27-s + 1.35·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.10·3-s + 0.5·4-s + 1.74·5-s + 0.777·6-s + 0.510·7-s − 0.353·8-s + 0.210·9-s − 1.23·10-s + 1.24·11-s − 0.550·12-s + 0.174·13-s − 0.361·14-s − 1.92·15-s + 0.250·16-s + 1.42·17-s − 0.148·18-s + 1.40·19-s + 0.874·20-s − 0.561·21-s − 0.879·22-s + 0.208·23-s + 0.388·24-s + 2.05·25-s − 0.123·26-s + 0.868·27-s + 0.255·28-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

\(L(1)\) \(\approx\) \(1.928437604\)
\(L(\frac12)\) \(\approx\) \(1.928437604\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.90T + 3T^{2} \)
5 \( 1 - 3.90T + 5T^{2} \)
7 \( 1 - 1.35T + 7T^{2} \)
11 \( 1 - 4.12T + 11T^{2} \)
13 \( 1 - 0.627T + 13T^{2} \)
17 \( 1 - 5.89T + 17T^{2} \)
19 \( 1 - 6.12T + 19T^{2} \)
29 \( 1 + 2.99T + 29T^{2} \)
31 \( 1 + 0.570T + 31T^{2} \)
37 \( 1 - 5.63T + 37T^{2} \)
41 \( 1 + 0.123T + 41T^{2} \)
43 \( 1 - 9.95T + 43T^{2} \)
47 \( 1 + 2.45T + 47T^{2} \)
53 \( 1 + 1.69T + 53T^{2} \)
59 \( 1 - 0.881T + 59T^{2} \)
61 \( 1 - 3.92T + 61T^{2} \)
67 \( 1 + 9.11T + 67T^{2} \)
71 \( 1 - 9.58T + 71T^{2} \)
73 \( 1 + 0.290T + 73T^{2} \)
79 \( 1 - 4.59T + 79T^{2} \)
83 \( 1 + 1.08T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−8.073334025417495350932411027567, −7.26617693428325563686499849860, −6.49921729576350466526711215980, −5.91933749749985997420363635266, −5.51700749158820391317370066590, −4.83585338711205399447115241599, −3.51020111296865506561910963377, −2.50389321020783001356289440870, −1.34824264193060318123650872936, −1.05823476267456939908517285336, 1.05823476267456939908517285336, 1.34824264193060318123650872936, 2.50389321020783001356289440870, 3.51020111296865506561910963377, 4.83585338711205399447115241599, 5.51700749158820391317370066590, 5.91933749749985997420363635266, 6.49921729576350466526711215980, 7.26617693428325563686499849860, 8.073334025417495350932411027567

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