Properties

Label 2-230-1.1-c3-0-16
Degree $2$
Conductor $230$
Sign $1$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 8.92·3-s + 4·4-s − 5·5-s + 17.8·6-s + 23.5·7-s + 8·8-s + 52.7·9-s − 10·10-s + 1.63·11-s + 35.7·12-s − 88.6·13-s + 47.1·14-s − 44.6·15-s + 16·16-s − 7.31·17-s + 105.·18-s − 6.53·19-s − 20·20-s + 210.·21-s + 3.27·22-s + 23·23-s + 71.4·24-s + 25·25-s − 177.·26-s + 229.·27-s + 94.2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.71·3-s + 0.5·4-s − 0.447·5-s + 1.21·6-s + 1.27·7-s + 0.353·8-s + 1.95·9-s − 0.316·10-s + 0.0448·11-s + 0.859·12-s − 1.89·13-s + 0.899·14-s − 0.768·15-s + 0.250·16-s − 0.104·17-s + 1.38·18-s − 0.0789·19-s − 0.223·20-s + 2.18·21-s + 0.0316·22-s + 0.208·23-s + 0.607·24-s + 0.200·25-s − 1.33·26-s + 1.63·27-s + 0.635·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.716132253\)
\(L(\frac12)\) \(\approx\) \(4.716132253\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
5 \( 1 + 5T \)
23 \( 1 - 23T \)
good3 \( 1 - 8.92T + 27T^{2} \)
7 \( 1 - 23.5T + 343T^{2} \)
11 \( 1 - 1.63T + 1.33e3T^{2} \)
13 \( 1 + 88.6T + 2.19e3T^{2} \)
17 \( 1 + 7.31T + 4.91e3T^{2} \)
19 \( 1 + 6.53T + 6.85e3T^{2} \)
29 \( 1 - 119.T + 2.43e4T^{2} \)
31 \( 1 + 156.T + 2.97e4T^{2} \)
37 \( 1 + 293.T + 5.06e4T^{2} \)
41 \( 1 - 74.3T + 6.89e4T^{2} \)
43 \( 1 - 468.T + 7.95e4T^{2} \)
47 \( 1 + 393.T + 1.03e5T^{2} \)
53 \( 1 - 233.T + 1.48e5T^{2} \)
59 \( 1 + 766.T + 2.05e5T^{2} \)
61 \( 1 - 178.T + 2.26e5T^{2} \)
67 \( 1 - 246.T + 3.00e5T^{2} \)
71 \( 1 + 650.T + 3.57e5T^{2} \)
73 \( 1 - 695.T + 3.89e5T^{2} \)
79 \( 1 - 660.T + 4.93e5T^{2} \)
83 \( 1 + 1.32e3T + 5.71e5T^{2} \)
89 \( 1 + 824.T + 7.04e5T^{2} \)
97 \( 1 - 383.T + 9.12e5T^{2} \)
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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

−12.04899734650032231224440300505, −10.85508026848754348364847229614, −9.685042683356738689637119605035, −8.612880457287322246343734423696, −7.72764494713986537189133122686, −7.13817456767071836379134482158, −5.04514031914002366085434525712, −4.20802066450498989423814407225, −2.89640745608174341720521656897, −1.87860545020693535218493606153, 1.87860545020693535218493606153, 2.89640745608174341720521656897, 4.20802066450498989423814407225, 5.04514031914002366085434525712, 7.13817456767071836379134482158, 7.72764494713986537189133122686, 8.612880457287322246343734423696, 9.685042683356738689637119605035, 10.85508026848754348364847229614, 12.04899734650032231224440300505

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