Properties

Label 2-1-1.1-c95-0-0
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $57.1535$
Root an. cond. $7.55999$
Motivic weight $95$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.34e14·2-s − 5.48e22·3-s + 1.54e28·4-s − 2.52e33·5-s − 1.28e37·6-s + 4.80e39·7-s − 5.67e42·8-s + 8.86e44·9-s − 5.92e47·10-s − 3.16e49·11-s − 8.45e50·12-s − 9.97e52·13-s + 1.12e54·14-s + 1.38e56·15-s − 1.94e57·16-s − 3.48e57·17-s + 2.07e59·18-s − 1.04e61·19-s − 3.89e61·20-s − 2.63e62·21-s − 7.41e63·22-s + 1.70e64·23-s + 3.11e65·24-s + 3.85e66·25-s − 2.34e67·26-s + 6.77e67·27-s + 7.41e67·28-s + ⋯
L(s)  = 1  + 1.17·2-s − 1.19·3-s + 0.389·4-s − 1.58·5-s − 1.40·6-s + 0.346·7-s − 0.719·8-s + 0.417·9-s − 1.87·10-s − 1.08·11-s − 0.463·12-s − 1.22·13-s + 0.408·14-s + 1.89·15-s − 1.23·16-s − 0.124·17-s + 0.492·18-s − 1.90·19-s − 0.618·20-s − 0.412·21-s − 1.27·22-s + 0.354·23-s + 0.857·24-s + 1.52·25-s − 1.43·26-s + 0.693·27-s + 0.134·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(96-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+95/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(57.1535\)
Root analytic conductor: \(7.55999\)
Motivic weight: \(95\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :95/2),\ 1)\)

Particular Values

\(L(48)\) \(\approx\) \(0.1932272041\)
\(L(\frac12)\) \(\approx\) \(0.1932272041\)
\(L(\frac{97}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2.34e14T + 3.96e28T^{2} \)
3 \( 1 + 5.48e22T + 2.12e45T^{2} \)
5 \( 1 + 2.52e33T + 2.52e66T^{2} \)
7 \( 1 - 4.80e39T + 1.92e80T^{2} \)
11 \( 1 + 3.16e49T + 8.55e98T^{2} \)
13 \( 1 + 9.97e52T + 6.67e105T^{2} \)
17 \( 1 + 3.48e57T + 7.80e116T^{2} \)
19 \( 1 + 1.04e61T + 3.03e121T^{2} \)
23 \( 1 - 1.70e64T + 2.31e129T^{2} \)
29 \( 1 - 1.55e69T + 8.46e138T^{2} \)
31 \( 1 - 1.02e71T + 4.77e141T^{2} \)
37 \( 1 + 2.81e74T + 9.53e148T^{2} \)
41 \( 1 - 3.26e76T + 1.63e153T^{2} \)
43 \( 1 - 7.50e75T + 1.51e155T^{2} \)
47 \( 1 + 2.42e79T + 7.06e158T^{2} \)
53 \( 1 + 1.40e82T + 6.40e163T^{2} \)
59 \( 1 + 1.17e84T + 1.70e168T^{2} \)
61 \( 1 - 7.58e84T + 4.03e169T^{2} \)
67 \( 1 + 8.84e86T + 2.99e173T^{2} \)
71 \( 1 - 6.46e86T + 7.40e175T^{2} \)
73 \( 1 - 1.01e88T + 1.03e177T^{2} \)
79 \( 1 - 1.04e89T + 1.88e180T^{2} \)
83 \( 1 - 1.05e91T + 2.05e182T^{2} \)
89 \( 1 + 3.90e92T + 1.55e185T^{2} \)
97 \( 1 - 3.16e94T + 5.53e188T^{2} \)
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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

−14.76558482114116219395317720943, −12.70344250977504814559515652710, −11.91035969877517106035673115176, −10.80054139244034980642929981896, −8.148916410262129479383055613495, −6.54767206257223383814843548490, −4.97665394737848794352314258144, −4.43825486666419652326591175600, −2.81807133490234266071961439540, −0.19972503428601819353330564652, 0.19972503428601819353330564652, 2.81807133490234266071961439540, 4.43825486666419652326591175600, 4.97665394737848794352314258144, 6.54767206257223383814843548490, 8.148916410262129479383055613495, 10.80054139244034980642929981896, 11.91035969877517106035673115176, 12.70344250977504814559515652710, 14.76558482114116219395317720943

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