Properties

Label 2-6034-1.1-c1-0-104
Degree $2$
Conductor $6034$
Sign $1$
Analytic cond. $48.1817$
Root an. cond. $6.94130$
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 + 1.23·3-s + 4-s + 0.483·5-s + 1.23·6-s + 7-s + 8-s − 1.47·9-s + 0.483·10-s − 0.207·11-s + 1.23·12-s + 4.81·13-s + 14-s + 0.597·15-s + 16-s + 6.97·17-s − 1.47·18-s − 3.74·19-s + 0.483·20-s + 1.23·21-s − 0.207·22-s − 2.03·23-s + 1.23·24-s − 4.76·25-s + 4.81·26-s − 5.52·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.713·3-s + 0.5·4-s + 0.216·5-s + 0.504·6-s + 0.377·7-s + 0.353·8-s − 0.491·9-s + 0.152·10-s − 0.0625·11-s + 0.356·12-s + 1.33·13-s + 0.267·14-s + 0.154·15-s + 0.250·16-s + 1.69·17-s − 0.347·18-s − 0.859·19-s + 0.108·20-s + 0.269·21-s − 0.0442·22-s − 0.425·23-s + 0.252·24-s − 0.953·25-s + 0.943·26-s − 1.06·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6034\)    =    \(2 \cdot 7 \cdot 431\)
Sign: $1$
Analytic conductor: \(48.1817\)
Root analytic conductor: \(6.94130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.844708826\)
\(L(\frac12)\) \(\approx\) \(4.844708826\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
431 \( 1 - T \)
good3 \( 1 - 1.23T + 3T^{2} \)
5 \( 1 - 0.483T + 5T^{2} \)
11 \( 1 + 0.207T + 11T^{2} \)
13 \( 1 - 4.81T + 13T^{2} \)
17 \( 1 - 6.97T + 17T^{2} \)
19 \( 1 + 3.74T + 19T^{2} \)
23 \( 1 + 2.03T + 23T^{2} \)
29 \( 1 + 1.08T + 29T^{2} \)
31 \( 1 - 7.69T + 31T^{2} \)
37 \( 1 - 4.45T + 37T^{2} \)
41 \( 1 + 2.15T + 41T^{2} \)
43 \( 1 - 5.26T + 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + 3.34T + 53T^{2} \)
59 \( 1 + 1.64T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 4.48T + 73T^{2} \)
79 \( 1 + 7.86T + 79T^{2} \)
83 \( 1 - 4.48T + 83T^{2} \)
89 \( 1 + 5.10T + 89T^{2} \)
97 \( 1 - 9.35T + 97T^{2} \)
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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

−8.031797759485074756201990189859, −7.57451719380198446746520859349, −6.39832531473143324830490016122, −5.89438005941209945095084835176, −5.33791924861222469729561314223, −4.19063398320255715165170532600, −3.70576042925809841533237168172, −2.86050409789472359050594530468, −2.09979804589005090759587802197, −1.06034089898346341506337680806, 1.06034089898346341506337680806, 2.09979804589005090759587802197, 2.86050409789472359050594530468, 3.70576042925809841533237168172, 4.19063398320255715165170532600, 5.33791924861222469729561314223, 5.89438005941209945095084835176, 6.39832531473143324830490016122, 7.57451719380198446746520859349, 8.031797759485074756201990189859

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