Properties

Label 2-230-1.1-c1-0-3
Degree $2$
Conductor $230$
Sign $1$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.79·3-s + 4-s − 5-s − 1.79·6-s + 2.79·7-s − 8-s + 0.208·9-s + 10-s + 3.79·11-s + 1.79·12-s + 1.20·13-s − 2.79·14-s − 1.79·15-s + 16-s − 3.79·17-s − 0.208·18-s + 1.20·19-s − 20-s + 5·21-s − 3.79·22-s + 23-s − 1.79·24-s + 25-s − 1.20·26-s − 5.00·27-s + 2.79·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.03·3-s + 0.5·4-s − 0.447·5-s − 0.731·6-s + 1.05·7-s − 0.353·8-s + 0.0695·9-s + 0.316·10-s + 1.14·11-s + 0.517·12-s + 0.335·13-s − 0.746·14-s − 0.462·15-s + 0.250·16-s − 0.919·17-s − 0.0491·18-s + 0.277·19-s − 0.223·20-s + 1.09·21-s − 0.808·22-s + 0.208·23-s − 0.365·24-s + 0.200·25-s − 0.237·26-s − 0.962·27-s + 0.527·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.272117732\)
\(L(\frac12)\) \(\approx\) \(1.272117732\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 - 1.79T + 3T^{2} \)
7 \( 1 - 2.79T + 7T^{2} \)
11 \( 1 - 3.79T + 11T^{2} \)
13 \( 1 - 1.20T + 13T^{2} \)
17 \( 1 + 3.79T + 17T^{2} \)
19 \( 1 - 1.20T + 19T^{2} \)
29 \( 1 + 1.58T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 2.20T + 41T^{2} \)
43 \( 1 + 7.16T + 43T^{2} \)
47 \( 1 + 13.5T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 4.41T + 59T^{2} \)
61 \( 1 + 3.37T + 61T^{2} \)
67 \( 1 + 7.16T + 67T^{2} \)
71 \( 1 + 5.37T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 3.16T + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−11.74263410731868075600580313835, −11.41971376748791719919366959666, −10.10507068480568508870022571273, −8.891536565685023601694878685968, −8.501411878293945551716222971506, −7.57143152749856172316050467033, −6.41055578761420862138865627363, −4.60961578868387518096880916533, −3.25470319294818186689101430882, −1.69968985125155088025702730602, 1.69968985125155088025702730602, 3.25470319294818186689101430882, 4.60961578868387518096880916533, 6.41055578761420862138865627363, 7.57143152749856172316050467033, 8.501411878293945551716222971506, 8.891536565685023601694878685968, 10.10507068480568508870022571273, 11.41971376748791719919366959666, 11.74263410731868075600580313835

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