Properties

Label 2-230-1.1-c3-0-3
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 + 0.589·3-s + 4·4-s − 5·5-s − 1.17·6-s − 18.5·7-s − 8·8-s − 26.6·9-s + 10·10-s + 47.9·11-s + 2.35·12-s + 42.3·13-s + 37.0·14-s − 2.94·15-s + 16·16-s + 1.70·17-s + 53.3·18-s + 21.4·19-s − 20·20-s − 10.9·21-s − 95.8·22-s − 23·23-s − 4.71·24-s + 25·25-s − 84.7·26-s − 31.6·27-s − 74.0·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.113·3-s + 0.5·4-s − 0.447·5-s − 0.0802·6-s − 0.999·7-s − 0.353·8-s − 0.987·9-s + 0.316·10-s + 1.31·11-s + 0.0567·12-s + 0.903·13-s + 0.706·14-s − 0.0507·15-s + 0.250·16-s + 0.0243·17-s + 0.697·18-s + 0.258·19-s − 0.223·20-s − 0.113·21-s − 0.928·22-s − 0.208·23-s − 0.0401·24-s + 0.200·25-s − 0.639·26-s − 0.225·27-s − 0.499·28-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: $\chi_{230} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.047376531\)
\(L(\frac12)\) \(\approx\) \(1.047376531\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.589T + 27T^{2} \)
7 \( 1 + 18.5T + 343T^{2} \)
11 \( 1 - 47.9T + 1.33e3T^{2} \)
13 \( 1 - 42.3T + 2.19e3T^{2} \)
17 \( 1 - 1.70T + 4.91e3T^{2} \)
19 \( 1 - 21.4T + 6.85e3T^{2} \)
29 \( 1 - 57.6T + 2.43e4T^{2} \)
31 \( 1 - 295.T + 2.97e4T^{2} \)
37 \( 1 + 7.85T + 5.06e4T^{2} \)
41 \( 1 - 465.T + 6.89e4T^{2} \)
43 \( 1 - 182.T + 7.95e4T^{2} \)
47 \( 1 - 449.T + 1.03e5T^{2} \)
53 \( 1 + 368.T + 1.48e5T^{2} \)
59 \( 1 + 377.T + 2.05e5T^{2} \)
61 \( 1 - 849.T + 2.26e5T^{2} \)
67 \( 1 - 92.3T + 3.00e5T^{2} \)
71 \( 1 + 626.T + 3.57e5T^{2} \)
73 \( 1 - 439.T + 3.89e5T^{2} \)
79 \( 1 - 641.T + 4.93e5T^{2} \)
83 \( 1 + 609.T + 5.71e5T^{2} \)
89 \( 1 - 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 1.42e3T + 9.12e5T^{2} \)
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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

−11.65836451076832049138511530701, −10.81505583792425751126455375858, −9.588277649748678898892067227271, −8.901443017497042459704749871343, −7.987328015995729355076230620203, −6.66228505681297785097783241915, −5.96938972579766822693923211827, −3.97657288497232002418100685636, −2.85032150788752427345480824765, −0.841714331942519847881041423366, 0.841714331942519847881041423366, 2.85032150788752427345480824765, 3.97657288497232002418100685636, 5.96938972579766822693923211827, 6.66228505681297785097783241915, 7.987328015995729355076230620203, 8.901443017497042459704749871343, 9.588277649748678898892067227271, 10.81505583792425751126455375858, 11.65836451076832049138511530701

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