Properties

Label 2-1-1.1-c49-0-1
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $15.2066$
Root an. cond. $3.89956$
Motivic weight $49$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.25e7·2-s + 5.57e11·3-s − 5.34e13·4-s − 1.06e17·5-s − 1.25e19·6-s + 5.68e20·7-s + 1.39e22·8-s + 7.20e22·9-s + 2.40e24·10-s − 4.66e25·11-s − 2.98e25·12-s + 1.95e27·13-s − 1.28e28·14-s − 5.94e28·15-s − 2.83e29·16-s − 1.59e30·17-s − 1.62e30·18-s − 3.45e31·19-s + 5.69e30·20-s + 3.16e32·21-s + 1.05e33·22-s − 2.46e33·23-s + 7.76e33·24-s − 6.42e33·25-s − 4.41e34·26-s − 9.33e34·27-s − 3.03e34·28-s + ⋯
L(s)  = 1  − 0.951·2-s + 1.14·3-s − 0.0949·4-s − 0.798·5-s − 1.08·6-s + 1.12·7-s + 1.04·8-s + 0.301·9-s + 0.759·10-s − 1.42·11-s − 0.108·12-s + 0.998·13-s − 1.06·14-s − 0.911·15-s − 0.896·16-s − 1.13·17-s − 0.286·18-s − 1.61·19-s + 0.0758·20-s + 1.27·21-s + 1.35·22-s − 1.07·23-s + 1.18·24-s − 0.361·25-s − 0.949·26-s − 0.797·27-s − 0.106·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(50-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+49/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(15.2066\)
Root analytic conductor: \(3.89956\)
Motivic weight: \(49\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :49/2),\ -1)\)

Particular Values

\(L(25)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{51}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 2.25e7T + 5.62e14T^{2} \)
3 \( 1 - 5.57e11T + 2.39e23T^{2} \)
5 \( 1 + 1.06e17T + 1.77e34T^{2} \)
7 \( 1 - 5.68e20T + 2.56e41T^{2} \)
11 \( 1 + 4.66e25T + 1.06e51T^{2} \)
13 \( 1 - 1.95e27T + 3.83e54T^{2} \)
17 \( 1 + 1.59e30T + 1.95e60T^{2} \)
19 \( 1 + 3.45e31T + 4.55e62T^{2} \)
23 \( 1 + 2.46e33T + 5.30e66T^{2} \)
29 \( 1 - 4.27e35T + 4.54e71T^{2} \)
31 \( 1 - 2.26e36T + 1.19e73T^{2} \)
37 \( 1 + 1.55e38T + 6.94e76T^{2} \)
41 \( 1 + 1.49e39T + 1.06e79T^{2} \)
43 \( 1 - 5.17e39T + 1.09e80T^{2} \)
47 \( 1 - 5.79e40T + 8.56e81T^{2} \)
53 \( 1 + 5.51e41T + 3.08e84T^{2} \)
59 \( 1 + 1.93e42T + 5.91e86T^{2} \)
61 \( 1 - 2.41e43T + 3.02e87T^{2} \)
67 \( 1 + 8.25e44T + 3.00e89T^{2} \)
71 \( 1 + 1.77e45T + 5.14e90T^{2} \)
73 \( 1 - 6.25e45T + 2.00e91T^{2} \)
79 \( 1 - 4.72e46T + 9.63e92T^{2} \)
83 \( 1 - 4.29e46T + 1.08e94T^{2} \)
89 \( 1 + 7.42e47T + 3.31e95T^{2} \)
97 \( 1 + 5.90e48T + 2.24e97T^{2} \)
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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

−19.34464022492593974823149617727, −17.90380556562652169931080960624, −15.49229808217256898622175699422, −13.64577534776753551998143639650, −10.78289761962258789745604528082, −8.511098314252081387804742690107, −8.008874204200169355909432833896, −4.27851438882376458693410009900, −2.06803731890084326163634144588, 0, 2.06803731890084326163634144588, 4.27851438882376458693410009900, 8.008874204200169355909432833896, 8.511098314252081387804742690107, 10.78289761962258789745604528082, 13.64577534776753551998143639650, 15.49229808217256898622175699422, 17.90380556562652169931080960624, 19.34464022492593974823149617727

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