Properties

Label 2-230-1.1-c5-0-27
Degree $2$
Conductor $230$
Sign $-1$
Analytic cond. $36.8882$
Root an. cond. $6.07357$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·3-s + 16·4-s − 25·5-s − 32·6-s + 199·7-s − 64·8-s − 179·9-s + 100·10-s + 150·11-s + 128·12-s − 1.20e3·13-s − 796·14-s − 200·15-s + 256·16-s + 735·17-s + 716·18-s − 22·19-s − 400·20-s + 1.59e3·21-s − 600·22-s − 529·23-s − 512·24-s + 625·25-s + 4.80e3·26-s − 3.37e3·27-s + 3.18e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.513·3-s + 1/2·4-s − 0.447·5-s − 0.362·6-s + 1.53·7-s − 0.353·8-s − 0.736·9-s + 0.316·10-s + 0.373·11-s + 0.256·12-s − 1.97·13-s − 1.08·14-s − 0.229·15-s + 1/4·16-s + 0.616·17-s + 0.520·18-s − 0.0139·19-s − 0.223·20-s + 0.787·21-s − 0.264·22-s − 0.208·23-s − 0.181·24-s + 1/5·25-s + 1.39·26-s − 0.891·27-s + 0.767·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(36.8882\)
Root analytic conductor: \(6.07357\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{230} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ -1)\)

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)$
bad2 \( 1 + p^{2} T \)
5 \( 1 + p^{2} T \)
23 \( 1 + p^{2} T \)
good3 \( 1 - 8 T + p^{5} T^{2} \)
7 \( 1 - 199 T + p^{5} T^{2} \)
11 \( 1 - 150 T + p^{5} T^{2} \)
13 \( 1 + 1202 T + p^{5} T^{2} \)
17 \( 1 - 735 T + p^{5} T^{2} \)
19 \( 1 + 22 T + p^{5} T^{2} \)
29 \( 1 + 5525 T + p^{5} T^{2} \)
31 \( 1 + 95 T + p^{5} T^{2} \)
37 \( 1 + 397 T + p^{5} T^{2} \)
41 \( 1 - 20633 T + p^{5} T^{2} \)
43 \( 1 + 11384 T + p^{5} T^{2} \)
47 \( 1 - 1992 T + p^{5} T^{2} \)
53 \( 1 + 7349 T + p^{5} T^{2} \)
59 \( 1 + 23827 T + p^{5} T^{2} \)
61 \( 1 + 44016 T + p^{5} T^{2} \)
67 \( 1 + 37713 T + p^{5} T^{2} \)
71 \( 1 + 50057 T + p^{5} T^{2} \)
73 \( 1 + 16698 T + p^{5} T^{2} \)
79 \( 1 + 31004 T + p^{5} T^{2} \)
83 \( 1 + 70077 T + p^{5} T^{2} \)
89 \( 1 - 7676 T + p^{5} T^{2} \)
97 \( 1 + 150094 T + p^{5} T^{2} \)
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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

−10.92478931862965906928124621387, −9.695287932727095481350942959946, −8.815280041525245543339260129928, −7.79645185320190228978750908186, −7.43650054493850525090907900767, −5.62837358611476366990358242167, −4.46315318006215781964724633901, −2.83223059447641587728913485020, −1.67496418093733385427697234596, 0, 1.67496418093733385427697234596, 2.83223059447641587728913485020, 4.46315318006215781964724633901, 5.62837358611476366990358242167, 7.43650054493850525090907900767, 7.79645185320190228978750908186, 8.815280041525245543339260129928, 9.695287932727095481350942959946, 10.92478931862965906928124621387

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