Properties

Label 2-23-1.1-c5-0-7
Degree $2$
Conductor $23$
Sign $-1$
Analytic cond. $3.68882$
Root an. cond. $1.92063$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.164·2-s − 1.43·3-s − 31.9·4-s − 37.9·5-s + 0.235·6-s − 43.8·7-s + 10.5·8-s − 240.·9-s + 6.24·10-s − 163.·11-s + 45.7·12-s + 430.·13-s + 7.21·14-s + 54.3·15-s + 1.02e3·16-s + 740.·17-s + 39.6·18-s − 916.·19-s + 1.21e3·20-s + 62.7·21-s + 26.8·22-s − 529·23-s − 15.0·24-s − 1.68e3·25-s − 70.7·26-s + 692.·27-s + 1.40e3·28-s + ⋯
L(s)  = 1  − 0.0290·2-s − 0.0917·3-s − 0.999·4-s − 0.679·5-s + 0.00266·6-s − 0.338·7-s + 0.0581·8-s − 0.991·9-s + 0.0197·10-s − 0.406·11-s + 0.0917·12-s + 0.706·13-s + 0.00983·14-s + 0.0623·15-s + 0.997·16-s + 0.621·17-s + 0.0288·18-s − 0.582·19-s + 0.678·20-s + 0.0310·21-s + 0.0118·22-s − 0.208·23-s − 0.00533·24-s − 0.538·25-s − 0.0205·26-s + 0.182·27-s + 0.337·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-1$
Analytic conductor: \(3.68882\)
Root analytic conductor: \(1.92063\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 23,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + 529T \)
good2 \( 1 + 0.164T + 32T^{2} \)
3 \( 1 + 1.43T + 243T^{2} \)
5 \( 1 + 37.9T + 3.12e3T^{2} \)
7 \( 1 + 43.8T + 1.68e4T^{2} \)
11 \( 1 + 163.T + 1.61e5T^{2} \)
13 \( 1 - 430.T + 3.71e5T^{2} \)
17 \( 1 - 740.T + 1.41e6T^{2} \)
19 \( 1 + 916.T + 2.47e6T^{2} \)
29 \( 1 + 5.11e3T + 2.05e7T^{2} \)
31 \( 1 + 5.70e3T + 2.86e7T^{2} \)
37 \( 1 + 1.09e4T + 6.93e7T^{2} \)
41 \( 1 - 1.10e4T + 1.15e8T^{2} \)
43 \( 1 - 5.52e3T + 1.47e8T^{2} \)
47 \( 1 - 1.44e4T + 2.29e8T^{2} \)
53 \( 1 - 6.85e3T + 4.18e8T^{2} \)
59 \( 1 - 4.15e3T + 7.14e8T^{2} \)
61 \( 1 + 2.19e4T + 8.44e8T^{2} \)
67 \( 1 + 1.51e4T + 1.35e9T^{2} \)
71 \( 1 - 1.40e4T + 1.80e9T^{2} \)
73 \( 1 + 2.33e4T + 2.07e9T^{2} \)
79 \( 1 - 6.42e4T + 3.07e9T^{2} \)
83 \( 1 - 1.14e5T + 3.93e9T^{2} \)
89 \( 1 + 6.34e4T + 5.58e9T^{2} \)
97 \( 1 + 9.19e4T + 8.58e9T^{2} \)
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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

−16.28130488900852758725213997978, −14.82994923139838765169869162226, −13.61784875165198485825219203222, −12.34386286650814691422551255976, −10.83507658233044476865893056574, −9.144695026783762343558798641703, −7.926176168376805307324222843550, −5.63300935067018173421652806968, −3.71137790019432399643875222060, 0, 3.71137790019432399643875222060, 5.63300935067018173421652806968, 7.926176168376805307324222843550, 9.144695026783762343558798641703, 10.83507658233044476865893056574, 12.34386286650814691422551255976, 13.61784875165198485825219203222, 14.82994923139838765169869162226, 16.28130488900852758725213997978

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