Properties

Label 2-1305-1.1-c1-0-37
Degree $2$
Conductor $1305$
Sign $1$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
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  + 2.79·2-s + 5.79·4-s − 5-s + 7-s + 10.5·8-s − 2.79·10-s − 5·11-s + 4.58·13-s + 2.79·14-s + 17.9·16-s + 3·17-s − 5.58·19-s − 5.79·20-s − 13.9·22-s + 4·23-s + 25-s + 12.7·26-s + 5.79·28-s − 29-s + 4·31-s + 28.9·32-s + 8.37·34-s − 35-s − 4·37-s − 15.5·38-s − 10.5·40-s − 9.16·41-s + ⋯
L(s)  = 1  + 1.97·2-s + 2.89·4-s − 0.447·5-s + 0.377·7-s + 3.74·8-s − 0.882·10-s − 1.50·11-s + 1.27·13-s + 0.746·14-s + 4.48·16-s + 0.727·17-s − 1.28·19-s − 1.29·20-s − 2.97·22-s + 0.834·23-s + 0.200·25-s + 2.50·26-s + 1.09·28-s − 0.185·29-s + 0.718·31-s + 5.11·32-s + 1.43·34-s − 0.169·35-s − 0.657·37-s − 2.52·38-s − 1.67·40-s − 1.43·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.664668808\)
\(L(\frac12)\) \(\approx\) \(5.664668808\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
29 \( 1 + T \)
good2 \( 1 - 2.79T + 2T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 - 4.58T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 5.58T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 9.16T + 41T^{2} \)
43 \( 1 + 0.417T + 43T^{2} \)
47 \( 1 + 1.41T + 47T^{2} \)
53 \( 1 + 9.58T + 53T^{2} \)
59 \( 1 + 1.58T + 59T^{2} \)
61 \( 1 + 14.7T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 - 0.417T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 1.58T + 79T^{2} \)
83 \( 1 - 2.41T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 2.41T + 97T^{2} \)
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

−10.15711805068897051049620413457, −8.357115316915182064836072936196, −7.87076498499061829011102609335, −6.85875844297782352079929851335, −6.11364945338429630048357510247, −5.20906301146983166694475064986, −4.64022693040538000446964690507, −3.59341397086996229110883014634, −2.89471569634958048054299088791, −1.66875754246049497979161296314, 1.66875754246049497979161296314, 2.89471569634958048054299088791, 3.59341397086996229110883014634, 4.64022693040538000446964690507, 5.20906301146983166694475064986, 6.11364945338429630048357510247, 6.85875844297782352079929851335, 7.87076498499061829011102609335, 8.357115316915182064836072936196, 10.15711805068897051049620413457

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