Properties

Label 2-1-1.1-c85-0-0
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $45.7549$
Root an. cond. $6.76424$
Motivic weight $85$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.05e13·2-s − 2.03e20·3-s + 7.20e25·4-s − 7.53e29·5-s + 2.13e33·6-s − 3.48e35·7-s − 3.51e38·8-s + 5.29e39·9-s + 7.93e42·10-s − 3.29e44·11-s − 1.46e46·12-s − 2.08e47·13-s + 3.66e48·14-s + 1.53e50·15-s + 9.12e50·16-s + 1.04e52·17-s − 5.57e52·18-s + 4.21e54·19-s − 5.43e55·20-s + 7.06e55·21-s + 3.46e57·22-s − 7.42e56·23-s + 7.13e58·24-s + 3.09e59·25-s + 2.19e60·26-s + 6.21e60·27-s − 2.51e61·28-s + ⋯
L(s)  = 1  − 1.69·2-s − 1.07·3-s + 1.86·4-s − 1.48·5-s + 1.81·6-s − 0.421·7-s − 1.46·8-s + 0.147·9-s + 2.50·10-s − 1.81·11-s − 1.99·12-s − 0.945·13-s + 0.713·14-s + 1.58·15-s + 0.609·16-s + 0.529·17-s − 0.249·18-s + 1.89·19-s − 2.76·20-s + 0.451·21-s + 3.06·22-s − 0.0994·23-s + 1.56·24-s + 1.19·25-s + 1.59·26-s + 0.913·27-s − 0.786·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(86-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+85/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(45.7549\)
Root analytic conductor: \(6.76424\)
Motivic weight: \(85\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :85/2),\ -1)\)

Particular Values

\(L(43)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{87}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.05e13T + 3.86e25T^{2} \)
3 \( 1 + 2.03e20T + 3.59e40T^{2} \)
5 \( 1 + 7.53e29T + 2.58e59T^{2} \)
7 \( 1 + 3.48e35T + 6.81e71T^{2} \)
11 \( 1 + 3.29e44T + 3.29e88T^{2} \)
13 \( 1 + 2.08e47T + 4.84e94T^{2} \)
17 \( 1 - 1.04e52T + 3.87e104T^{2} \)
19 \( 1 - 4.21e54T + 4.94e108T^{2} \)
23 \( 1 + 7.42e56T + 5.58e115T^{2} \)
29 \( 1 + 3.37e61T + 2.01e124T^{2} \)
31 \( 1 - 4.51e62T + 5.83e126T^{2} \)
37 \( 1 + 9.60e65T + 1.98e133T^{2} \)
41 \( 1 + 1.35e68T + 1.22e137T^{2} \)
43 \( 1 - 2.53e69T + 6.99e138T^{2} \)
47 \( 1 + 1.21e70T + 1.34e142T^{2} \)
53 \( 1 - 6.11e72T + 3.65e146T^{2} \)
59 \( 1 + 5.92e74T + 3.32e150T^{2} \)
61 \( 1 - 1.02e76T + 5.66e151T^{2} \)
67 \( 1 - 5.44e76T + 1.64e155T^{2} \)
71 \( 1 + 2.00e78T + 2.27e157T^{2} \)
73 \( 1 - 2.30e79T + 2.41e158T^{2} \)
79 \( 1 + 4.82e80T + 1.98e161T^{2} \)
83 \( 1 - 2.52e81T + 1.32e163T^{2} \)
89 \( 1 + 2.61e82T + 4.99e165T^{2} \)
97 \( 1 - 4.38e83T + 7.50e168T^{2} \)
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

−15.70979358715136013819785606027, −12.16126337221765337476078780992, −11.14185737996004624157094939808, −9.942132515370786841203146847347, −8.015604230534603784526369480716, −7.23408742363977038108993832760, −5.23446479268528100305459211684, −2.89027261378791911420091177585, −0.69538035289095819575750220929, 0, 0.69538035289095819575750220929, 2.89027261378791911420091177585, 5.23446479268528100305459211684, 7.23408742363977038108993832760, 8.015604230534603784526369480716, 9.942132515370786841203146847347, 11.14185737996004624157094939808, 12.16126337221765337476078780992, 15.70979358715136013819785606027

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