Properties

Label 2-1305-1.1-c1-0-23
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  + 2.16·2-s + 2.70·4-s − 5-s − 0.484·7-s + 1.53·8-s − 2.16·10-s + 2.30·11-s + 6.79·13-s − 1.05·14-s − 2.08·16-s + 4.79·17-s + 3.23·19-s − 2.70·20-s + 4.99·22-s − 3.55·23-s + 25-s + 14.7·26-s − 1.31·28-s + 29-s + 0.766·31-s − 7.59·32-s + 10.4·34-s + 0.484·35-s + 0.474·37-s + 7.01·38-s − 1.53·40-s − 2.45·41-s + ⋯
L(s)  = 1  + 1.53·2-s + 1.35·4-s − 0.447·5-s − 0.183·7-s + 0.541·8-s − 0.686·10-s + 0.694·11-s + 1.88·13-s − 0.281·14-s − 0.522·16-s + 1.16·17-s + 0.741·19-s − 0.605·20-s + 1.06·22-s − 0.741·23-s + 0.200·25-s + 2.89·26-s − 0.248·28-s + 0.185·29-s + 0.137·31-s − 1.34·32-s + 1.78·34-s + 0.0819·35-s + 0.0780·37-s + 1.13·38-s − 0.242·40-s − 0.382·41-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

\(L(1)\) \(\approx\) \(3.891029877\)
\(L(\frac12)\) \(\approx\) \(3.891029877\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.16T + 2T^{2} \)
7 \( 1 + 0.484T + 7T^{2} \)
11 \( 1 - 2.30T + 11T^{2} \)
13 \( 1 - 6.79T + 13T^{2} \)
17 \( 1 - 4.79T + 17T^{2} \)
19 \( 1 - 3.23T + 19T^{2} \)
23 \( 1 + 3.55T + 23T^{2} \)
31 \( 1 - 0.766T + 31T^{2} \)
37 \( 1 - 0.474T + 37T^{2} \)
41 \( 1 + 2.45T + 41T^{2} \)
43 \( 1 - 9.42T + 43T^{2} \)
47 \( 1 - 3.60T + 47T^{2} \)
53 \( 1 + 4.67T + 53T^{2} \)
59 \( 1 - 4.96T + 59T^{2} \)
61 \( 1 + 6.75T + 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 + 2.46T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 1.37T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 - 1.22T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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

−9.693419951447816285344888343644, −8.760788005879715105925591614920, −7.88074912968272688264716550144, −6.85934161344938490725156322531, −6.05649059370524482846098749418, −5.51537702191694207560922168045, −4.29465463002541131876381669413, −3.70237346913367078031509253429, −2.98815246567583561681664837086, −1.32865209688490805257829647409, 1.32865209688490805257829647409, 2.98815246567583561681664837086, 3.70237346913367078031509253429, 4.29465463002541131876381669413, 5.51537702191694207560922168045, 6.05649059370524482846098749418, 6.85934161344938490725156322531, 7.88074912968272688264716550144, 8.760788005879715105925591614920, 9.693419951447816285344888343644

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