Properties

Label 2-1589-1.1-c1-0-46
Degree $2$
Conductor $1589$
Sign $1$
Analytic cond. $12.6882$
Root an. cond. $3.56205$
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  + 2.81·2-s − 2.61·3-s + 5.90·4-s − 2.68·5-s − 7.34·6-s − 7-s + 10.9·8-s + 3.82·9-s − 7.55·10-s − 1.00·11-s − 15.4·12-s + 3.31·13-s − 2.81·14-s + 7.01·15-s + 19.0·16-s + 1.27·17-s + 10.7·18-s + 5.00·19-s − 15.8·20-s + 2.61·21-s − 2.81·22-s + 2.00·23-s − 28.6·24-s + 2.21·25-s + 9.32·26-s − 2.16·27-s − 5.90·28-s + ⋯
L(s)  = 1  + 1.98·2-s − 1.50·3-s + 2.95·4-s − 1.20·5-s − 2.99·6-s − 0.377·7-s + 3.88·8-s + 1.27·9-s − 2.38·10-s − 0.302·11-s − 4.45·12-s + 0.920·13-s − 0.751·14-s + 1.81·15-s + 4.76·16-s + 0.309·17-s + 2.53·18-s + 1.14·19-s − 3.54·20-s + 0.570·21-s − 0.600·22-s + 0.417·23-s − 5.85·24-s + 0.442·25-s + 1.82·26-s − 0.417·27-s − 1.11·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1589 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1589 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1589\)    =    \(7 \cdot 227\)
Sign: $1$
Analytic conductor: \(12.6882\)
Root analytic conductor: \(3.56205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1589,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.288136613\)
\(L(\frac12)\) \(\approx\) \(3.288136613\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
227 \( 1 - T \)
good2 \( 1 - 2.81T + 2T^{2} \)
3 \( 1 + 2.61T + 3T^{2} \)
5 \( 1 + 2.68T + 5T^{2} \)
11 \( 1 + 1.00T + 11T^{2} \)
13 \( 1 - 3.31T + 13T^{2} \)
17 \( 1 - 1.27T + 17T^{2} \)
19 \( 1 - 5.00T + 19T^{2} \)
23 \( 1 - 2.00T + 23T^{2} \)
29 \( 1 + 8.62T + 29T^{2} \)
31 \( 1 - 7.37T + 31T^{2} \)
37 \( 1 - 4.68T + 37T^{2} \)
41 \( 1 - 3.36T + 41T^{2} \)
43 \( 1 - 0.959T + 43T^{2} \)
47 \( 1 + 3.84T + 47T^{2} \)
53 \( 1 - 14.0T + 53T^{2} \)
59 \( 1 - 2.75T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + 1.72T + 67T^{2} \)
71 \( 1 + 6.80T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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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

−9.938211024530731180944161935871, −8.113359624068797797593182561098, −7.27940557601921122885100043111, −6.73525221550194110925327715550, −5.73049421873220176553785737967, −5.45971160180282726409600756807, −4.38298294796064086522027924748, −3.82287029676935466008535778379, −2.87330925787185815930017800457, −1.08260262670122920767143002215, 1.08260262670122920767143002215, 2.87330925787185815930017800457, 3.82287029676935466008535778379, 4.38298294796064086522027924748, 5.45971160180282726409600756807, 5.73049421873220176553785737967, 6.73525221550194110925327715550, 7.27940557601921122885100043111, 8.113359624068797797593182561098, 9.938211024530731180944161935871

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