Properties

Label 2-3850-1.1-c1-0-44
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s − 7-s − 8-s + 9-s − 11-s − 2·12-s + 4.74·13-s + 14-s + 16-s − 4.74·17-s − 18-s + 4.74·19-s + 2·21-s + 22-s − 4.74·23-s + 2·24-s − 4.74·26-s + 4·27-s − 28-s − 2.74·29-s − 6.74·31-s − 32-s + 2·33-s + 4.74·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 0.5·4-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.301·11-s − 0.577·12-s + 1.31·13-s + 0.267·14-s + 0.250·16-s − 1.15·17-s − 0.235·18-s + 1.08·19-s + 0.436·21-s + 0.213·22-s − 0.989·23-s + 0.408·24-s − 0.930·26-s + 0.769·27-s − 0.188·28-s − 0.509·29-s − 1.21·31-s − 0.176·32-s + 0.348·33-s + 0.813·34-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: \(1\)
Selberg data: \((2,\ 3850,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 3T^{2} \)
13 \( 1 - 4.74T + 13T^{2} \)
17 \( 1 + 4.74T + 17T^{2} \)
19 \( 1 - 4.74T + 19T^{2} \)
23 \( 1 + 4.74T + 23T^{2} \)
29 \( 1 + 2.74T + 29T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 - 1.25T + 53T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 0.744T + 73T^{2} \)
79 \( 1 + 4.74T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + 7.48T + 89T^{2} \)
97 \( 1 - 5.25T + 97T^{2} \)
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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

−8.219258908242235018153596843961, −7.27217043039939613556152074881, −6.63515838341925719101763811778, −5.87122519362157433728008435182, −5.50369414958207086046131236699, −4.32367678329535912781877119078, −3.42421131782095103206799609304, −2.26677785319632327953878773890, −1.06979406516936859324046319691, 0, 1.06979406516936859324046319691, 2.26677785319632327953878773890, 3.42421131782095103206799609304, 4.32367678329535912781877119078, 5.50369414958207086046131236699, 5.87122519362157433728008435182, 6.63515838341925719101763811778, 7.27217043039939613556152074881, 8.219258908242235018153596843961

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