Properties

Label 2-6026-1.1-c1-0-126
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 − 1.52·3-s + 4-s + 1.80·5-s − 1.52·6-s + 2.28·7-s + 8-s − 0.665·9-s + 1.80·10-s + 6.02·11-s − 1.52·12-s − 1.69·13-s + 2.28·14-s − 2.76·15-s + 16-s + 4.34·17-s − 0.665·18-s + 7.78·19-s + 1.80·20-s − 3.48·21-s + 6.02·22-s − 23-s − 1.52·24-s − 1.72·25-s − 1.69·26-s + 5.60·27-s + 2.28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.882·3-s + 0.5·4-s + 0.809·5-s − 0.623·6-s + 0.862·7-s + 0.353·8-s − 0.221·9-s + 0.572·10-s + 1.81·11-s − 0.441·12-s − 0.470·13-s + 0.609·14-s − 0.714·15-s + 0.250·16-s + 1.05·17-s − 0.156·18-s + 1.78·19-s + 0.404·20-s − 0.760·21-s + 1.28·22-s − 0.208·23-s − 0.311·24-s − 0.344·25-s − 0.332·26-s + 1.07·27-s + 0.431·28-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

\(L(1)\) \(\approx\) \(3.683207228\)
\(L(\frac12)\) \(\approx\) \(3.683207228\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.52T + 3T^{2} \)
5 \( 1 - 1.80T + 5T^{2} \)
7 \( 1 - 2.28T + 7T^{2} \)
11 \( 1 - 6.02T + 11T^{2} \)
13 \( 1 + 1.69T + 13T^{2} \)
17 \( 1 - 4.34T + 17T^{2} \)
19 \( 1 - 7.78T + 19T^{2} \)
29 \( 1 - 2.33T + 29T^{2} \)
31 \( 1 - 4.63T + 31T^{2} \)
37 \( 1 + 3.98T + 37T^{2} \)
41 \( 1 + 6.85T + 41T^{2} \)
43 \( 1 - 3.29T + 43T^{2} \)
47 \( 1 + 2.24T + 47T^{2} \)
53 \( 1 - 0.341T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 9.39T + 61T^{2} \)
67 \( 1 - 9.80T + 67T^{2} \)
71 \( 1 + 16.5T + 71T^{2} \)
73 \( 1 + 0.667T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + 1.05T + 89T^{2} \)
97 \( 1 + 8.59T + 97T^{2} \)
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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

−7.923540715587937494385848126466, −7.12185777491704402139732686979, −6.42184773392858986824826406100, −5.83999119597267055742191319307, −5.24231887032783260173552807589, −4.73006735350275286900056699355, −3.72401197162816680849785177894, −2.90197735405548377545279558369, −1.66548493548728353119942155721, −1.07267455446090966587748279701, 1.07267455446090966587748279701, 1.66548493548728353119942155721, 2.90197735405548377545279558369, 3.72401197162816680849785177894, 4.73006735350275286900056699355, 5.24231887032783260173552807589, 5.83999119597267055742191319307, 6.42184773392858986824826406100, 7.12185777491704402139732686979, 7.923540715587937494385848126466

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