Properties

Label 2-1045-1.1-c1-0-15
Degree $2$
Conductor $1045$
Sign $1$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
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  + 0.719·2-s + 0.163·3-s − 1.48·4-s − 5-s + 0.117·6-s + 1.05·7-s − 2.50·8-s − 2.97·9-s − 0.719·10-s + 11-s − 0.241·12-s + 3.88·13-s + 0.759·14-s − 0.163·15-s + 1.16·16-s + 5.65·17-s − 2.13·18-s − 19-s + 1.48·20-s + 0.172·21-s + 0.719·22-s + 2.78·23-s − 0.408·24-s + 25-s + 2.79·26-s − 0.974·27-s − 1.56·28-s + ⋯
L(s)  = 1  + 0.508·2-s + 0.0941·3-s − 0.741·4-s − 0.447·5-s + 0.0478·6-s + 0.399·7-s − 0.885·8-s − 0.991·9-s − 0.227·10-s + 0.301·11-s − 0.0698·12-s + 1.07·13-s + 0.203·14-s − 0.0421·15-s + 0.291·16-s + 1.37·17-s − 0.503·18-s − 0.229·19-s + 0.331·20-s + 0.0376·21-s + 0.153·22-s + 0.581·23-s − 0.0833·24-s + 0.200·25-s + 0.548·26-s − 0.187·27-s − 0.296·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.627474129\)
\(L(\frac12)\) \(\approx\) \(1.627474129\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 - 0.719T + 2T^{2} \)
3 \( 1 - 0.163T + 3T^{2} \)
7 \( 1 - 1.05T + 7T^{2} \)
13 \( 1 - 3.88T + 13T^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
23 \( 1 - 2.78T + 23T^{2} \)
29 \( 1 - 0.890T + 29T^{2} \)
31 \( 1 - 6.13T + 31T^{2} \)
37 \( 1 - 3.51T + 37T^{2} \)
41 \( 1 + 2.28T + 41T^{2} \)
43 \( 1 - 3.91T + 43T^{2} \)
47 \( 1 - 7.75T + 47T^{2} \)
53 \( 1 - 7.61T + 53T^{2} \)
59 \( 1 + 2.17T + 59T^{2} \)
61 \( 1 - 0.475T + 61T^{2} \)
67 \( 1 + 0.383T + 67T^{2} \)
71 \( 1 + 2.51T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 + 4.27T + 79T^{2} \)
83 \( 1 + 1.39T + 83T^{2} \)
89 \( 1 - 8.36T + 89T^{2} \)
97 \( 1 + 6.75T + 97T^{2} \)
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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

−9.867769301179894077325770430087, −8.837534683888809995962899203660, −8.444996853437422368973885236484, −7.58799056096592976348482919199, −6.22561224170944557449585168385, −5.58952776399998786942484760081, −4.62359158142365317075068531266, −3.70216848163064425812907354282, −2.91090536982010738435443149877, −0.951375051304025515462569909213, 0.951375051304025515462569909213, 2.91090536982010738435443149877, 3.70216848163064425812907354282, 4.62359158142365317075068531266, 5.58952776399998786942484760081, 6.22561224170944557449585168385, 7.58799056096592976348482919199, 8.444996853437422368973885236484, 8.837534683888809995962899203660, 9.867769301179894077325770430087

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