Properties

Label 2-5070-1.1-c1-0-34
Degree $2$
Conductor $5070$
Sign $1$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s + 4.80·7-s − 8-s + 9-s + 10-s + 2.55·11-s + 12-s − 4.80·14-s − 15-s + 16-s − 3.02·17-s − 18-s − 3.89·19-s − 20-s + 4.80·21-s − 2.55·22-s − 1.64·23-s − 24-s + 25-s + 27-s + 4.80·28-s + 5.07·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.81·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.770·11-s + 0.288·12-s − 1.28·14-s − 0.258·15-s + 0.250·16-s − 0.734·17-s − 0.235·18-s − 0.894·19-s − 0.223·20-s + 1.04·21-s − 0.544·22-s − 0.342·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.907·28-s + 0.942·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.159290914\)
\(L(\frac12)\) \(\approx\) \(2.159290914\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 \)
good7 \( 1 - 4.80T + 7T^{2} \)
11 \( 1 - 2.55T + 11T^{2} \)
17 \( 1 + 3.02T + 17T^{2} \)
19 \( 1 + 3.89T + 19T^{2} \)
23 \( 1 + 1.64T + 23T^{2} \)
29 \( 1 - 5.07T + 29T^{2} \)
31 \( 1 - 5.44T + 31T^{2} \)
37 \( 1 - 7.51T + 37T^{2} \)
41 \( 1 + 5.89T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 6.42T + 47T^{2} \)
53 \( 1 + 6.91T + 53T^{2} \)
59 \( 1 - 7.60T + 59T^{2} \)
61 \( 1 - 9.65T + 61T^{2} \)
67 \( 1 - 0.929T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 + 5.40T + 79T^{2} \)
83 \( 1 - 4.33T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 - 17.9T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.381086587859211177207195728916, −7.85889169676724704710059292166, −6.90736348814798056870876331202, −6.40776626153954386104477563313, −5.14432823229774307438050967178, −4.45761261521035848546731509054, −3.81135494100607934206314630857, −2.53418287910930099257937495119, −1.84579632741894035076039638235, −0.912579351530342337874080420322, 0.912579351530342337874080420322, 1.84579632741894035076039638235, 2.53418287910930099257937495119, 3.81135494100607934206314630857, 4.45761261521035848546731509054, 5.14432823229774307438050967178, 6.40776626153954386104477563313, 6.90736348814798056870876331202, 7.85889169676724704710059292166, 8.381086587859211177207195728916

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