Properties

Label 2-1-1.1-c91-0-6
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $52.4421$
Root an. cond. $7.24169$
Motivic weight $91$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.15e13·2-s + 9.10e21·3-s + 2.64e27·4-s + 1.09e32·5-s + 6.51e35·6-s − 1.62e38·7-s + 1.22e40·8-s + 5.67e43·9-s + 7.84e45·10-s − 9.18e46·11-s + 2.41e49·12-s + 1.76e50·13-s − 1.16e52·14-s + 9.98e53·15-s − 5.67e54·16-s − 5.68e55·17-s + 4.06e57·18-s + 1.30e58·19-s + 2.90e59·20-s − 1.48e60·21-s − 6.57e60·22-s − 6.51e61·23-s + 1.11e62·24-s + 7.96e63·25-s + 1.26e64·26-s + 2.78e65·27-s − 4.30e65·28-s + ⋯
L(s)  = 1  + 1.43·2-s + 1.78·3-s + 1.06·4-s + 1.72·5-s + 2.56·6-s − 0.574·7-s + 0.0996·8-s + 2.16·9-s + 2.48·10-s − 0.380·11-s + 1.90·12-s + 0.364·13-s − 0.825·14-s + 3.06·15-s − 0.925·16-s − 0.587·17-s + 3.11·18-s + 0.854·19-s + 1.84·20-s − 1.02·21-s − 0.546·22-s − 0.716·23-s + 0.177·24-s + 1.97·25-s + 0.524·26-s + 2.07·27-s − 0.613·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(92-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+91/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(52.4421\)
Root analytic conductor: \(7.24169\)
Motivic weight: \(91\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :91/2),\ 1)\)

Particular Values

\(L(46)\) \(\approx\) \(10.43203425\)
\(L(\frac12)\) \(\approx\) \(10.43203425\)
\(L(\frac{93}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 7.15e13T + 2.47e27T^{2} \)
3 \( 1 - 9.10e21T + 2.61e43T^{2} \)
5 \( 1 - 1.09e32T + 4.03e63T^{2} \)
7 \( 1 + 1.62e38T + 8.01e76T^{2} \)
11 \( 1 + 9.18e46T + 5.84e94T^{2} \)
13 \( 1 - 1.76e50T + 2.33e101T^{2} \)
17 \( 1 + 5.68e55T + 9.35e111T^{2} \)
19 \( 1 - 1.30e58T + 2.32e116T^{2} \)
23 \( 1 + 6.51e61T + 8.26e123T^{2} \)
29 \( 1 - 7.19e65T + 1.19e133T^{2} \)
31 \( 1 + 6.19e67T + 5.17e135T^{2} \)
37 \( 1 + 3.48e71T + 5.08e142T^{2} \)
41 \( 1 + 4.92e72T + 5.79e146T^{2} \)
43 \( 1 + 1.54e74T + 4.42e148T^{2} \)
47 \( 1 - 9.93e75T + 1.44e152T^{2} \)
53 \( 1 - 3.16e78T + 8.11e156T^{2} \)
59 \( 1 + 1.69e80T + 1.40e161T^{2} \)
61 \( 1 - 2.96e81T + 2.91e162T^{2} \)
67 \( 1 - 8.96e82T + 1.48e166T^{2} \)
71 \( 1 - 1.16e84T + 2.91e168T^{2} \)
73 \( 1 - 4.43e84T + 3.65e169T^{2} \)
79 \( 1 + 3.89e86T + 4.83e172T^{2} \)
83 \( 1 - 1.58e87T + 4.32e174T^{2} \)
89 \( 1 + 7.73e88T + 2.48e177T^{2} \)
97 \( 1 - 1.03e90T + 6.25e180T^{2} \)
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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

−14.27430480430287381515511518664, −13.62987338284484517456086896022, −12.87598149103374013430117672418, −9.964151129888293410101646105562, −8.872361882508068627877283882524, −6.76584472535588108790046290612, −5.35928503236760057448960674999, −3.68974127345054066341664260043, −2.67530921326377898571649231972, −1.83364336925779462073382397785, 1.83364336925779462073382397785, 2.67530921326377898571649231972, 3.68974127345054066341664260043, 5.35928503236760057448960674999, 6.76584472535588108790046290612, 8.872361882508068627877283882524, 9.964151129888293410101646105562, 12.87598149103374013430117672418, 13.62987338284484517456086896022, 14.27430480430287381515511518664

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