Properties

Label 2-230-1.1-c3-0-0
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 − 7.57·3-s + 4·4-s − 5·5-s + 15.1·6-s − 35.4·7-s − 8·8-s + 30.3·9-s + 10·10-s − 16.6·11-s − 30.2·12-s − 79.9·13-s + 70.8·14-s + 37.8·15-s + 16·16-s − 46.8·17-s − 60.6·18-s − 110.·19-s − 20·20-s + 268.·21-s + 33.2·22-s − 23·23-s + 60.5·24-s + 25·25-s + 159.·26-s − 25.2·27-s − 141.·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.45·3-s + 0.5·4-s − 0.447·5-s + 1.03·6-s − 1.91·7-s − 0.353·8-s + 1.12·9-s + 0.316·10-s − 0.455·11-s − 0.728·12-s − 1.70·13-s + 1.35·14-s + 0.651·15-s + 0.250·16-s − 0.667·17-s − 0.794·18-s − 1.33·19-s − 0.223·20-s + 2.78·21-s + 0.322·22-s − 0.208·23-s + 0.515·24-s + 0.200·25-s + 1.20·26-s − 0.180·27-s − 0.956·28-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

\(L(2)\) \(\approx\) \(0.03207607858\)
\(L(\frac12)\) \(\approx\) \(0.03207607858\)
\(L(\frac{5}{2})\) 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 \( 1 + 7.57T + 27T^{2} \)
7 \( 1 + 35.4T + 343T^{2} \)
11 \( 1 + 16.6T + 1.33e3T^{2} \)
13 \( 1 + 79.9T + 2.19e3T^{2} \)
17 \( 1 + 46.8T + 4.91e3T^{2} \)
19 \( 1 + 110.T + 6.85e3T^{2} \)
29 \( 1 + 0.836T + 2.43e4T^{2} \)
31 \( 1 + 119.T + 2.97e4T^{2} \)
37 \( 1 - 368.T + 5.06e4T^{2} \)
41 \( 1 + 95.7T + 6.89e4T^{2} \)
43 \( 1 - 331.T + 7.95e4T^{2} \)
47 \( 1 + 535.T + 1.03e5T^{2} \)
53 \( 1 - 409.T + 1.48e5T^{2} \)
59 \( 1 + 352.T + 2.05e5T^{2} \)
61 \( 1 + 507.T + 2.26e5T^{2} \)
67 \( 1 + 820.T + 3.00e5T^{2} \)
71 \( 1 + 733.T + 3.57e5T^{2} \)
73 \( 1 + 91.4T + 3.89e5T^{2} \)
79 \( 1 - 329.T + 4.93e5T^{2} \)
83 \( 1 + 753.T + 5.71e5T^{2} \)
89 \( 1 + 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + 271.T + 9.12e5T^{2} \)
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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

−11.70522469191020168940755008056, −10.65060760472449553445011066582, −10.02248467551984072542645536989, −9.092825093184862820472203729348, −7.49205151837231804454471851439, −6.64892521124386147676795191987, −5.89214661485449211643328953521, −4.45471270430794561664189539621, −2.69184090756482823469013641294, −0.14443467134602199433264596732, 0.14443467134602199433264596732, 2.69184090756482823469013641294, 4.45471270430794561664189539621, 5.89214661485449211643328953521, 6.64892521124386147676795191987, 7.49205151837231804454471851439, 9.092825093184862820472203729348, 10.02248467551984072542645536989, 10.65060760472449553445011066582, 11.70522469191020168940755008056

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