Properties

Label 2-1-1.1-c63-0-4
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $25.1360$
Root an. cond. $5.01359$
Motivic weight $63$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.72e9·2-s + 8.02e14·3-s + 1.30e19·4-s + 1.36e22·5-s + 3.78e24·6-s + 4.00e26·7-s + 1.82e28·8-s − 5.00e29·9-s + 6.44e31·10-s − 8.56e32·11-s + 1.04e34·12-s + 2.07e35·13-s + 1.88e36·14-s + 1.09e37·15-s − 3.46e37·16-s − 5.37e38·17-s − 2.36e39·18-s − 4.25e39·19-s + 1.78e41·20-s + 3.21e41·21-s − 4.04e42·22-s + 7.22e42·23-s + 1.46e43·24-s + 7.77e43·25-s + 9.79e44·26-s − 1.32e45·27-s + 5.23e45·28-s + ⋯
L(s)  = 1  + 1.55·2-s + 0.749·3-s + 1.41·4-s + 1.31·5-s + 1.16·6-s + 0.958·7-s + 0.649·8-s − 0.437·9-s + 2.03·10-s − 1.34·11-s + 1.06·12-s + 1.68·13-s + 1.49·14-s + 0.982·15-s − 0.407·16-s − 0.935·17-s − 0.680·18-s − 0.222·19-s + 1.85·20-s + 0.718·21-s − 2.09·22-s + 0.921·23-s + 0.487·24-s + 0.716·25-s + 2.62·26-s − 1.07·27-s + 1.35·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(64-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+63/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(25.1360\)
Root analytic conductor: \(5.01359\)
Motivic weight: \(63\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :63/2),\ 1)\)

Particular Values

\(L(32)\) \(\approx\) \(7.105748185\)
\(L(\frac12)\) \(\approx\) \(7.105748185\)
\(L(\frac{65}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 4.72e9T + 9.22e18T^{2} \)
3 \( 1 - 8.02e14T + 1.14e30T^{2} \)
5 \( 1 - 1.36e22T + 1.08e44T^{2} \)
7 \( 1 - 4.00e26T + 1.74e53T^{2} \)
11 \( 1 + 8.56e32T + 4.05e65T^{2} \)
13 \( 1 - 2.07e35T + 1.50e70T^{2} \)
17 \( 1 + 5.37e38T + 3.29e77T^{2} \)
19 \( 1 + 4.25e39T + 3.64e80T^{2} \)
23 \( 1 - 7.22e42T + 6.14e85T^{2} \)
29 \( 1 - 6.34e45T + 1.35e92T^{2} \)
31 \( 1 - 1.09e47T + 9.03e93T^{2} \)
37 \( 1 + 2.36e49T + 6.26e98T^{2} \)
41 \( 1 - 2.67e50T + 4.03e101T^{2} \)
43 \( 1 + 2.12e51T + 8.10e102T^{2} \)
47 \( 1 + 3.01e52T + 2.19e105T^{2} \)
53 \( 1 + 1.69e52T + 4.25e108T^{2} \)
59 \( 1 + 7.40e54T + 3.66e111T^{2} \)
61 \( 1 + 2.87e56T + 2.99e112T^{2} \)
67 \( 1 + 2.05e57T + 1.10e115T^{2} \)
71 \( 1 - 2.18e58T + 4.25e116T^{2} \)
73 \( 1 + 1.91e58T + 2.45e117T^{2} \)
79 \( 1 + 1.43e59T + 3.55e119T^{2} \)
83 \( 1 - 2.96e60T + 7.97e120T^{2} \)
89 \( 1 + 3.32e60T + 6.47e122T^{2} \)
97 \( 1 + 5.80e61T + 1.46e125T^{2} \)
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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

−17.92695554615381251828472088503, −15.35221874678101662383990215569, −13.88608247779151394549217633363, −13.30796128553540754992967398102, −10.98637406823321537021352577259, −8.572137877718424829530249086383, −6.11308787721379987450502308199, −4.89862718473805193834153393419, −3.01726640481319240814008038374, −1.90718296516908783667460242487, 1.90718296516908783667460242487, 3.01726640481319240814008038374, 4.89862718473805193834153393419, 6.11308787721379987450502308199, 8.572137877718424829530249086383, 10.98637406823321537021352577259, 13.30796128553540754992967398102, 13.88608247779151394549217633363, 15.35221874678101662383990215569, 17.92695554615381251828472088503

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