Properties

Label 2-230-1.1-c3-0-2
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 − 4.74·3-s + 4·4-s − 5·5-s + 9.49·6-s + 29.3·7-s − 8·8-s − 4.44·9-s + 10·10-s − 38.1·11-s − 18.9·12-s − 22.5·13-s − 58.7·14-s + 23.7·15-s + 16·16-s − 104.·17-s + 8.89·18-s + 141.·19-s − 20·20-s − 139.·21-s + 76.3·22-s − 23·23-s + 37.9·24-s + 25·25-s + 45.0·26-s + 149.·27-s + 117.·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.913·3-s + 0.5·4-s − 0.447·5-s + 0.646·6-s + 1.58·7-s − 0.353·8-s − 0.164·9-s + 0.316·10-s − 1.04·11-s − 0.456·12-s − 0.480·13-s − 1.12·14-s + 0.408·15-s + 0.250·16-s − 1.48·17-s + 0.116·18-s + 1.71·19-s − 0.223·20-s − 1.44·21-s + 0.739·22-s − 0.208·23-s + 0.323·24-s + 0.200·25-s + 0.340·26-s + 1.06·27-s + 0.792·28-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

\(L(2)\) \(\approx\) \(0.7948871915\)
\(L(\frac12)\) \(\approx\) \(0.7948871915\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.74T + 27T^{2} \)
7 \( 1 - 29.3T + 343T^{2} \)
11 \( 1 + 38.1T + 1.33e3T^{2} \)
13 \( 1 + 22.5T + 2.19e3T^{2} \)
17 \( 1 + 104.T + 4.91e3T^{2} \)
19 \( 1 - 141.T + 6.85e3T^{2} \)
29 \( 1 - 241.T + 2.43e4T^{2} \)
31 \( 1 - 99.2T + 2.97e4T^{2} \)
37 \( 1 - 59.9T + 5.06e4T^{2} \)
41 \( 1 - 249.T + 6.89e4T^{2} \)
43 \( 1 + 163.T + 7.95e4T^{2} \)
47 \( 1 + 205.T + 1.03e5T^{2} \)
53 \( 1 - 491.T + 1.48e5T^{2} \)
59 \( 1 - 433.T + 2.05e5T^{2} \)
61 \( 1 - 660.T + 2.26e5T^{2} \)
67 \( 1 + 323.T + 3.00e5T^{2} \)
71 \( 1 - 893.T + 3.57e5T^{2} \)
73 \( 1 - 196.T + 3.89e5T^{2} \)
79 \( 1 + 500.T + 4.93e5T^{2} \)
83 \( 1 - 800.T + 5.71e5T^{2} \)
89 \( 1 + 729.T + 7.04e5T^{2} \)
97 \( 1 - 1.13e3T + 9.12e5T^{2} \)
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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

−11.49432267756180009994680997296, −11.00805353501760541392368963471, −10.03958151604533505639337895230, −8.574655855616886075106987483725, −7.916229632306941277190238852518, −6.87539608483631960585577945622, −5.42243924147952364428299570897, −4.68229888209149143825694265876, −2.51179737146138258283283578057, −0.76139953237893540068576956796, 0.76139953237893540068576956796, 2.51179737146138258283283578057, 4.68229888209149143825694265876, 5.42243924147952364428299570897, 6.87539608483631960585577945622, 7.916229632306941277190238852518, 8.574655855616886075106987483725, 10.03958151604533505639337895230, 11.00805353501760541392368963471, 11.49432267756180009994680997296

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