Properties

Label 2-1-1.1-c71-0-2
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $31.9246$
Root an. cond. $5.65018$
Motivic weight $71$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.24e10·2-s − 1.51e17·3-s + 2.88e21·4-s − 9.47e24·5-s − 1.09e28·6-s + 1.67e28·7-s + 3.77e31·8-s + 1.54e34·9-s − 6.86e35·10-s − 8.90e36·11-s − 4.36e38·12-s + 2.84e39·13-s + 1.21e39·14-s + 1.43e42·15-s − 4.07e42·16-s + 3.55e43·17-s + 1.11e45·18-s + 2.02e45·19-s − 2.73e46·20-s − 2.53e45·21-s − 6.45e47·22-s + 2.53e48·23-s − 5.72e48·24-s + 4.74e49·25-s + 2.05e50·26-s − 1.19e51·27-s + 4.81e49·28-s + ⋯
L(s)  = 1  + 1.49·2-s − 1.74·3-s + 1.22·4-s − 1.45·5-s − 2.60·6-s + 0.0166·7-s + 0.329·8-s + 2.05·9-s − 2.16·10-s − 0.955·11-s − 2.13·12-s + 0.810·13-s + 0.0248·14-s + 2.54·15-s − 0.730·16-s + 0.740·17-s + 3.06·18-s + 0.814·19-s − 1.77·20-s − 0.0291·21-s − 1.42·22-s + 1.15·23-s − 0.575·24-s + 1.11·25-s + 1.20·26-s − 1.84·27-s + 0.0203·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(72-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+71/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(31.9246\)
Root analytic conductor: \(5.65018\)
Motivic weight: \(71\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :71/2),\ 1)\)

Particular Values

\(L(36)\) \(\approx\) \(1.498955761\)
\(L(\frac12)\) \(\approx\) \(1.498955761\)
\(L(\frac{73}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 7.24e10T + 2.36e21T^{2} \)
3 \( 1 + 1.51e17T + 7.50e33T^{2} \)
5 \( 1 + 9.47e24T + 4.23e49T^{2} \)
7 \( 1 - 1.67e28T + 1.00e60T^{2} \)
11 \( 1 + 8.90e36T + 8.68e73T^{2} \)
13 \( 1 - 2.84e39T + 1.23e79T^{2} \)
17 \( 1 - 3.55e43T + 2.30e87T^{2} \)
19 \( 1 - 2.02e45T + 6.18e90T^{2} \)
23 \( 1 - 2.53e48T + 4.81e96T^{2} \)
29 \( 1 + 9.84e51T + 6.76e103T^{2} \)
31 \( 1 - 2.46e52T + 7.70e105T^{2} \)
37 \( 1 - 2.16e55T + 2.19e111T^{2} \)
41 \( 1 - 2.82e56T + 3.21e114T^{2} \)
43 \( 1 + 6.60e56T + 9.46e115T^{2} \)
47 \( 1 - 2.68e59T + 5.23e118T^{2} \)
53 \( 1 - 8.64e60T + 2.65e122T^{2} \)
59 \( 1 - 1.09e63T + 5.37e125T^{2} \)
61 \( 1 + 2.05e63T + 5.73e126T^{2} \)
67 \( 1 + 1.19e65T + 4.48e129T^{2} \)
71 \( 1 - 7.43e65T + 2.75e131T^{2} \)
73 \( 1 - 1.10e66T + 1.97e132T^{2} \)
79 \( 1 + 2.10e67T + 5.38e134T^{2} \)
83 \( 1 - 4.06e67T + 1.79e136T^{2} \)
89 \( 1 - 1.73e69T + 2.55e138T^{2} \)
97 \( 1 - 1.20e70T + 1.15e141T^{2} \)
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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

−16.30987238974797643578562870067, −15.35742699330843518638874664542, −12.96296489652118700889724575001, −11.83275875755529896998379297739, −10.99369808909137559980720845121, −7.32383613248478915281172666756, −5.75502253196001092456127417807, −4.74886003849360771254735722472, −3.48746634772373262871765394213, −0.65951141890227211426020692908, 0.65951141890227211426020692908, 3.48746634772373262871765394213, 4.74886003849360771254735722472, 5.75502253196001092456127417807, 7.32383613248478915281172666756, 10.99369808909137559980720845121, 11.83275875755529896998379297739, 12.96296489652118700889724575001, 15.35742699330843518638874664542, 16.30987238974797643578562870067

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