Properties

Label 2-845-1.1-c1-0-33
Degree $2$
Conductor $845$
Sign $1$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + 2.20·2-s − 0.0130·3-s + 2.86·4-s + 5-s − 0.0287·6-s + 4.60·7-s + 1.90·8-s − 2.99·9-s + 2.20·10-s + 2.93·11-s − 0.0374·12-s + 10.1·14-s − 0.0130·15-s − 1.52·16-s − 3.35·17-s − 6.61·18-s − 2.46·19-s + 2.86·20-s − 0.0601·21-s + 6.46·22-s − 1.58·23-s − 0.0248·24-s + 25-s + 0.0783·27-s + 13.2·28-s + 8.26·29-s − 0.0287·30-s + ⋯
L(s)  = 1  + 1.55·2-s − 0.00753·3-s + 1.43·4-s + 0.447·5-s − 0.0117·6-s + 1.74·7-s + 0.674·8-s − 0.999·9-s + 0.697·10-s + 0.884·11-s − 0.0107·12-s + 2.71·14-s − 0.00337·15-s − 0.380·16-s − 0.812·17-s − 1.55·18-s − 0.564·19-s + 0.640·20-s − 0.0131·21-s + 1.37·22-s − 0.330·23-s − 0.00508·24-s + 0.200·25-s + 0.0150·27-s + 2.49·28-s + 1.53·29-s − 0.00525·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.198768132\)
\(L(\frac12)\) \(\approx\) \(4.198768132\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - 2.20T + 2T^{2} \)
3 \( 1 + 0.0130T + 3T^{2} \)
7 \( 1 - 4.60T + 7T^{2} \)
11 \( 1 - 2.93T + 11T^{2} \)
17 \( 1 + 3.35T + 17T^{2} \)
19 \( 1 + 2.46T + 19T^{2} \)
23 \( 1 + 1.58T + 23T^{2} \)
29 \( 1 - 8.26T + 29T^{2} \)
31 \( 1 + 9.77T + 31T^{2} \)
37 \( 1 - 4.12T + 37T^{2} \)
41 \( 1 - 7.26T + 41T^{2} \)
43 \( 1 + 0.705T + 43T^{2} \)
47 \( 1 - 8.57T + 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + 5.33T + 59T^{2} \)
61 \( 1 - 1.92T + 61T^{2} \)
67 \( 1 + 7.29T + 67T^{2} \)
71 \( 1 + 6.68T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 0.984T + 79T^{2} \)
83 \( 1 + 7.84T + 83T^{2} \)
89 \( 1 + 0.412T + 89T^{2} \)
97 \( 1 + 3.38T + 97T^{2} \)
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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

−10.75301964506587892251325318945, −9.139277008791345559370143345175, −8.540278268536354129441400636564, −7.41851087010809377722999709623, −6.28831671144721963418674621111, −5.68522322418331732873168303865, −4.72376500877832604181603311727, −4.15839053742035583461779654970, −2.76333460841635509444102783460, −1.76654217246964337355133498607, 1.76654217246964337355133498607, 2.76333460841635509444102783460, 4.15839053742035583461779654970, 4.72376500877832604181603311727, 5.68522322418331732873168303865, 6.28831671144721963418674621111, 7.41851087010809377722999709623, 8.540278268536354129441400636564, 9.139277008791345559370143345175, 10.75301964506587892251325318945

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