Properties

Label 2-6026-1.1-c1-0-13
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 − 1.65·3-s + 4-s − 0.00351·5-s − 1.65·6-s − 4.74·7-s + 8-s − 0.255·9-s − 0.00351·10-s − 2.21·11-s − 1.65·12-s − 6.69·13-s − 4.74·14-s + 0.00583·15-s + 16-s − 1.31·17-s − 0.255·18-s + 5.56·19-s − 0.00351·20-s + 7.86·21-s − 2.21·22-s − 23-s − 1.65·24-s − 4.99·25-s − 6.69·26-s + 5.39·27-s − 4.74·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.956·3-s + 0.5·4-s − 0.00157·5-s − 0.676·6-s − 1.79·7-s + 0.353·8-s − 0.0852·9-s − 0.00111·10-s − 0.669·11-s − 0.478·12-s − 1.85·13-s − 1.26·14-s + 0.00150·15-s + 0.250·16-s − 0.318·17-s − 0.0602·18-s + 1.27·19-s − 0.000786·20-s + 1.71·21-s − 0.473·22-s − 0.208·23-s − 0.338·24-s − 0.999·25-s − 1.31·26-s + 1.03·27-s − 0.896·28-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

\(L(1)\) \(\approx\) \(0.3942925235\)
\(L(\frac12)\) \(\approx\) \(0.3942925235\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.65T + 3T^{2} \)
5 \( 1 + 0.00351T + 5T^{2} \)
7 \( 1 + 4.74T + 7T^{2} \)
11 \( 1 + 2.21T + 11T^{2} \)
13 \( 1 + 6.69T + 13T^{2} \)
17 \( 1 + 1.31T + 17T^{2} \)
19 \( 1 - 5.56T + 19T^{2} \)
29 \( 1 + 4.14T + 29T^{2} \)
31 \( 1 + 5.44T + 31T^{2} \)
37 \( 1 + 8.90T + 37T^{2} \)
41 \( 1 - 5.74T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 - 5.51T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + 8.33T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 - 3.28T + 71T^{2} \)
73 \( 1 - 5.89T + 73T^{2} \)
79 \( 1 - 3.48T + 79T^{2} \)
83 \( 1 + 0.374T + 83T^{2} \)
89 \( 1 + 8.56T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−7.58295016678363027908992773577, −7.26104110607588955041705703529, −6.52323490029489880621294292067, −5.74673742206242598905784234992, −5.39820881680305791798996698850, −4.64659923685448218420737061347, −3.54446318994318662044330739126, −2.97645934924245612189726461681, −2.10145539508028656579987210892, −0.28392858543827512758336623043, 0.28392858543827512758336623043, 2.10145539508028656579987210892, 2.97645934924245612189726461681, 3.54446318994318662044330739126, 4.64659923685448218420737061347, 5.39820881680305791798996698850, 5.74673742206242598905784234992, 6.52323490029489880621294292067, 7.26104110607588955041705703529, 7.58295016678363027908992773577

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