Properties

Label 2-1-1.1-c49-0-2
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  + 2.37e7·2-s − 2.01e10·3-s + 1.82e12·4-s + 2.93e15·5-s − 4.79e17·6-s − 2.24e20·7-s − 1.33e22·8-s − 2.38e23·9-s + 6.96e22·10-s + 2.83e25·11-s − 3.68e22·12-s − 1.95e27·13-s − 5.34e27·14-s − 5.91e25·15-s − 3.17e29·16-s − 2.51e30·17-s − 5.67e30·18-s + 1.78e30·19-s + 5.34e27·20-s + 4.53e30·21-s + 6.73e32·22-s + 3.03e33·23-s + 2.69e32·24-s − 1.77e34·25-s − 4.64e34·26-s + 9.65e33·27-s − 4.10e32·28-s + ⋯
L(s)  = 1  + 1.00·2-s − 0.0412·3-s + 0.00324·4-s + 0.0219·5-s − 0.0413·6-s − 0.443·7-s − 0.998·8-s − 0.998·9-s + 0.0220·10-s + 0.867·11-s − 0.000133·12-s − 0.998·13-s − 0.444·14-s − 0.000907·15-s − 1.00·16-s − 1.79·17-s − 0.999·18-s + 0.0837·19-s + 7.12e−5·20-s + 0.0183·21-s + 0.868·22-s + 1.31·23-s + 0.0411·24-s − 0.999·25-s − 1.00·26-s + 0.0824·27-s − 0.00143·28-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 \( 1 - 2.37e7T + 5.62e14T^{2} \)
3 \( 1 + 2.01e10T + 2.39e23T^{2} \)
5 \( 1 - 2.93e15T + 1.77e34T^{2} \)
7 \( 1 + 2.24e20T + 2.56e41T^{2} \)
11 \( 1 - 2.83e25T + 1.06e51T^{2} \)
13 \( 1 + 1.95e27T + 3.83e54T^{2} \)
17 \( 1 + 2.51e30T + 1.95e60T^{2} \)
19 \( 1 - 1.78e30T + 4.55e62T^{2} \)
23 \( 1 - 3.03e33T + 5.30e66T^{2} \)
29 \( 1 - 2.16e35T + 4.54e71T^{2} \)
31 \( 1 - 4.19e36T + 1.19e73T^{2} \)
37 \( 1 - 3.37e38T + 6.94e76T^{2} \)
41 \( 1 + 7.11e38T + 1.06e79T^{2} \)
43 \( 1 + 1.14e40T + 1.09e80T^{2} \)
47 \( 1 - 5.26e40T + 8.56e81T^{2} \)
53 \( 1 - 1.13e42T + 3.08e84T^{2} \)
59 \( 1 + 2.86e43T + 5.91e86T^{2} \)
61 \( 1 + 7.30e43T + 3.02e87T^{2} \)
67 \( 1 + 7.70e44T + 3.00e89T^{2} \)
71 \( 1 - 2.28e44T + 5.14e90T^{2} \)
73 \( 1 + 5.39e45T + 2.00e91T^{2} \)
79 \( 1 + 5.09e46T + 9.63e92T^{2} \)
83 \( 1 - 1.71e47T + 1.08e94T^{2} \)
89 \( 1 - 2.40e47T + 3.31e95T^{2} \)
97 \( 1 - 2.78e48T + 2.24e97T^{2} \)
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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

−19.76760520402153564888327622916, −17.36619134628449050722607351470, −14.99119836444912637958177980122, −13.48145910801877039197681375547, −11.75701241907280302598158799096, −9.099084253729332542690739709602, −6.33339507006627846871768884874, −4.58684981886443562953941294497, −2.82882696054372990146097340768, 0, 2.82882696054372990146097340768, 4.58684981886443562953941294497, 6.33339507006627846871768884874, 9.099084253729332542690739709602, 11.75701241907280302598158799096, 13.48145910801877039197681375547, 14.99119836444912637958177980122, 17.36619134628449050722607351470, 19.76760520402153564888327622916

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