Properties

Label 2-1150-1.1-c1-0-11
Degree $2$
Conductor $1150$
Sign $1$
Analytic cond. $9.18279$
Root an. cond. $3.03031$
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 + 0.706·3-s + 4-s + 0.706·6-s − 2.40·7-s + 8-s − 2.50·9-s + 4.95·11-s + 0.706·12-s + 4.40·13-s − 2.40·14-s + 16-s − 0.200·17-s − 2.50·18-s + 6.65·19-s − 1.70·21-s + 4.95·22-s + 23-s + 0.706·24-s + 4.40·26-s − 3.88·27-s − 2.40·28-s + 1.67·29-s − 1.82·31-s + 32-s + 3.50·33-s − 0.200·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.407·3-s + 0.5·4-s + 0.288·6-s − 0.910·7-s + 0.353·8-s − 0.833·9-s + 1.49·11-s + 0.203·12-s + 1.22·13-s − 0.643·14-s + 0.250·16-s − 0.0487·17-s − 0.589·18-s + 1.52·19-s − 0.371·21-s + 1.05·22-s + 0.208·23-s + 0.144·24-s + 0.864·26-s − 0.748·27-s − 0.455·28-s + 0.311·29-s − 0.327·31-s + 0.176·32-s + 0.609·33-s − 0.0344·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(9.18279\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.891408604\)
\(L(\frac12)\) \(\approx\) \(2.891408604\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 - T \)
good3 \( 1 - 0.706T + 3T^{2} \)
7 \( 1 + 2.40T + 7T^{2} \)
11 \( 1 - 4.95T + 11T^{2} \)
13 \( 1 - 4.40T + 13T^{2} \)
17 \( 1 + 0.200T + 17T^{2} \)
19 \( 1 - 6.65T + 19T^{2} \)
29 \( 1 - 1.67T + 29T^{2} \)
31 \( 1 + 1.82T + 31T^{2} \)
37 \( 1 - 8.47T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 2.06T + 43T^{2} \)
47 \( 1 + 0.505T + 47T^{2} \)
53 \( 1 + 5.84T + 53T^{2} \)
59 \( 1 + 4.41T + 59T^{2} \)
61 \( 1 + 6.67T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 - 3.61T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 - 9.95T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 5.04T + 89T^{2} \)
97 \( 1 + 3.79T + 97T^{2} \)
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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

−9.481745513631826084891734137914, −9.212002281946319828218329999616, −8.146383851894313246654591852957, −7.16807273780554223021201214583, −6.18136965873921016182702735043, −5.83733296624171783265238845461, −4.39901784012987546315663699004, −3.47626542437341727155408012057, −2.91055280880067350922482125998, −1.27211587416571166300629431866, 1.27211587416571166300629431866, 2.91055280880067350922482125998, 3.47626542437341727155408012057, 4.39901784012987546315663699004, 5.83733296624171783265238845461, 6.18136965873921016182702735043, 7.16807273780554223021201214583, 8.146383851894313246654591852957, 9.212002281946319828218329999616, 9.481745513631826084891734137914

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