Properties

Label 2-3850-1.1-c1-0-46
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 + 3.27·3-s + 4-s − 3.27·6-s + 7-s − 8-s + 7.72·9-s + 11-s + 3.27·12-s − 6.46·13-s − 14-s + 16-s − 3.38·17-s − 7.72·18-s + 6.62·19-s + 3.27·21-s − 22-s + 5.53·23-s − 3.27·24-s + 6.46·26-s + 15.4·27-s + 28-s − 1.01·29-s − 2.72·31-s − 32-s + 3.27·33-s + 3.38·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.89·3-s + 0.5·4-s − 1.33·6-s + 0.377·7-s − 0.353·8-s + 2.57·9-s + 0.301·11-s + 0.945·12-s − 1.79·13-s − 0.267·14-s + 0.250·16-s − 0.820·17-s − 1.82·18-s + 1.51·19-s + 0.714·21-s − 0.213·22-s + 1.15·23-s − 0.668·24-s + 1.26·26-s + 2.97·27-s + 0.188·28-s − 0.188·29-s − 0.489·31-s − 0.176·32-s + 0.570·33-s + 0.579·34-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

\(L(1)\) \(\approx\) \(3.047996371\)
\(L(\frac12)\) \(\approx\) \(3.047996371\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.27T + 3T^{2} \)
13 \( 1 + 6.46T + 13T^{2} \)
17 \( 1 + 3.38T + 17T^{2} \)
19 \( 1 - 6.62T + 19T^{2} \)
23 \( 1 - 5.53T + 23T^{2} \)
29 \( 1 + 1.01T + 29T^{2} \)
31 \( 1 + 2.72T + 31T^{2} \)
37 \( 1 + 2.15T + 37T^{2} \)
41 \( 1 - 2.81T + 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 8.39T + 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 5.64T + 61T^{2} \)
67 \( 1 + 3.16T + 67T^{2} \)
71 \( 1 - 4.85T + 71T^{2} \)
73 \( 1 + 0.831T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 4.81T + 83T^{2} \)
89 \( 1 - 8.02T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−8.681177535416154116586351189339, −7.70750262916381002348733224527, −7.33948158038544420518909465775, −6.89169831827544393796290458431, −5.37161117650915665771209262398, −4.52174169930076183380655727940, −3.60035999110064682928895223265, −2.66473927516429406839083687662, −2.21909447721546870588093376138, −1.07763915755655351865225135593, 1.07763915755655351865225135593, 2.21909447721546870588093376138, 2.66473927516429406839083687662, 3.60035999110064682928895223265, 4.52174169930076183380655727940, 5.37161117650915665771209262398, 6.89169831827544393796290458431, 7.33948158038544420518909465775, 7.70750262916381002348733224527, 8.681177535416154116586351189339

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