Properties

Label 2-3850-1.1-c1-0-33
Degree $2$
Conductor $3850$
Sign $1$
Analytic cond. $30.7424$
Root an. cond. $5.54458$
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 + 0.529·3-s + 4-s + 0.529·6-s − 7-s + 8-s − 2.71·9-s + 11-s + 0.529·12-s + 2.71·13-s − 14-s + 16-s + 2.52·17-s − 2.71·18-s − 1.77·19-s − 0.529·21-s + 22-s − 6.27·23-s + 0.529·24-s + 2.71·26-s − 3.02·27-s − 28-s + 7.33·29-s + 5.36·31-s + 32-s + 0.529·33-s + 2.52·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.305·3-s + 0.5·4-s + 0.216·6-s − 0.377·7-s + 0.353·8-s − 0.906·9-s + 0.301·11-s + 0.152·12-s + 0.754·13-s − 0.267·14-s + 0.250·16-s + 0.613·17-s − 0.641·18-s − 0.408·19-s − 0.115·21-s + 0.213·22-s − 1.30·23-s + 0.108·24-s + 0.533·26-s − 0.582·27-s − 0.188·28-s + 1.36·29-s + 0.963·31-s + 0.176·32-s + 0.0921·33-s + 0.433·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3850\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(30.7424\)
Root analytic conductor: \(5.54458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.226607649\)
\(L(\frac12)\) \(\approx\) \(3.226607649\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good3 \( 1 - 0.529T + 3T^{2} \)
13 \( 1 - 2.71T + 13T^{2} \)
17 \( 1 - 2.52T + 17T^{2} \)
19 \( 1 + 1.77T + 19T^{2} \)
23 \( 1 + 6.27T + 23T^{2} \)
29 \( 1 - 7.33T + 29T^{2} \)
31 \( 1 - 5.36T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 - 8.99T + 41T^{2} \)
43 \( 1 - 1.94T + 43T^{2} \)
47 \( 1 + 1.96T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + 7.02T + 59T^{2} \)
61 \( 1 - 0.443T + 61T^{2} \)
67 \( 1 + 0.941T + 67T^{2} \)
71 \( 1 - 1.16T + 71T^{2} \)
73 \( 1 - 2.29T + 73T^{2} \)
79 \( 1 + 3.22T + 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 + 9.77T + 89T^{2} \)
97 \( 1 - 1.16T + 97T^{2} \)
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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

−8.261214927446294291817606481258, −7.935540767523949312469357375097, −6.80542571288583062425101106816, −6.03911210518406873040814796746, −5.76304636250686467061952429203, −4.51284439204362721226014059822, −3.90242813840368687485290448795, −2.98971071466105910403455534270, −2.34422788019405519528201713629, −0.933053065227256125194141120400, 0.933053065227256125194141120400, 2.34422788019405519528201713629, 2.98971071466105910403455534270, 3.90242813840368687485290448795, 4.51284439204362721226014059822, 5.76304636250686467061952429203, 6.03911210518406873040814796746, 6.80542571288583062425101106816, 7.935540767523949312469357375097, 8.261214927446294291817606481258

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