Properties

Label 2-1-1.1-c87-0-3
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $47.9333$
Root an. cond. $6.92339$
Motivic weight $87$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.89e13·2-s + 2.95e20·3-s + 2.06e26·4-s − 5.49e29·5-s − 5.61e33·6-s − 2.67e35·7-s − 9.77e38·8-s − 2.35e41·9-s + 1.04e43·10-s + 2.95e44·11-s + 6.09e46·12-s + 5.04e48·13-s + 5.08e48·14-s − 1.62e50·15-s − 1.33e52·16-s + 3.69e53·17-s + 4.48e54·18-s − 3.90e55·19-s − 1.13e56·20-s − 7.91e55·21-s − 5.61e57·22-s − 2.02e59·23-s − 2.89e59·24-s − 6.16e60·25-s − 9.58e61·26-s − 1.65e62·27-s − 5.52e61·28-s + ⋯
L(s)  = 1  − 1.52·2-s + 0.519·3-s + 1.33·4-s − 0.216·5-s − 0.794·6-s − 0.0463·7-s − 0.508·8-s − 0.729·9-s + 0.330·10-s + 0.147·11-s + 0.692·12-s + 1.76·13-s + 0.0707·14-s − 0.112·15-s − 0.556·16-s + 1.10·17-s + 1.11·18-s − 0.923·19-s − 0.288·20-s − 0.0240·21-s − 0.225·22-s − 1.17·23-s − 0.264·24-s − 0.953·25-s − 2.69·26-s − 0.899·27-s − 0.0617·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(88-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+87/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(47.9333\)
Root analytic conductor: \(6.92339\)
Motivic weight: \(87\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :87/2),\ 1)\)

Particular Values

\(L(44)\) \(\approx\) \(0.9508247917\)
\(L(\frac12)\) \(\approx\) \(0.9508247917\)
\(L(\frac{89}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.89e13T + 1.54e26T^{2} \)
3 \( 1 - 2.95e20T + 3.23e41T^{2} \)
5 \( 1 + 5.49e29T + 6.46e60T^{2} \)
7 \( 1 + 2.67e35T + 3.33e73T^{2} \)
11 \( 1 - 2.95e44T + 3.99e90T^{2} \)
13 \( 1 - 5.04e48T + 8.18e96T^{2} \)
17 \( 1 - 3.69e53T + 1.11e107T^{2} \)
19 \( 1 + 3.90e55T + 1.78e111T^{2} \)
23 \( 1 + 2.02e59T + 2.95e118T^{2} \)
29 \( 1 - 5.13e62T + 1.69e127T^{2} \)
31 \( 1 - 8.31e64T + 5.60e129T^{2} \)
37 \( 1 + 2.25e67T + 2.71e136T^{2} \)
41 \( 1 + 1.59e70T + 2.05e140T^{2} \)
43 \( 1 + 2.55e70T + 1.29e142T^{2} \)
47 \( 1 - 7.76e72T + 2.96e145T^{2} \)
53 \( 1 - 7.96e74T + 1.02e150T^{2} \)
59 \( 1 - 1.11e77T + 1.15e154T^{2} \)
61 \( 1 + 3.76e77T + 2.10e155T^{2} \)
67 \( 1 - 4.73e79T + 7.38e158T^{2} \)
71 \( 1 + 7.60e79T + 1.14e161T^{2} \)
73 \( 1 - 1.77e81T + 1.28e162T^{2} \)
79 \( 1 + 3.09e82T + 1.24e165T^{2} \)
83 \( 1 - 1.37e83T + 9.11e166T^{2} \)
89 \( 1 - 2.89e84T + 3.95e169T^{2} \)
97 \( 1 - 4.83e86T + 7.06e172T^{2} \)
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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

−15.71530848136916740393583130877, −13.83925096881549102667384680875, −11.55024120100984511769055529769, −10.15102423508026145447206161885, −8.681954242032503208388519347381, −7.994757496284795804620330574646, −6.17694126477491749490990848106, −3.65347321122838347249598890075, −1.99766475071213553700683468274, −0.68104828465265253401117786909, 0.68104828465265253401117786909, 1.99766475071213553700683468274, 3.65347321122838347249598890075, 6.17694126477491749490990848106, 7.994757496284795804620330574646, 8.681954242032503208388519347381, 10.15102423508026145447206161885, 11.55024120100984511769055529769, 13.83925096881549102667384680875, 15.71530848136916740393583130877

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