Properties

Label 2-230-23.22-c4-0-13
Degree $2$
Conductor $230$
Sign $0.932 - 0.362i$
Analytic cond. $23.7750$
Root an. cond. $4.87597$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·2-s − 11.4·3-s + 8.00·4-s + 11.1i·5-s − 32.3·6-s − 10.3i·7-s + 22.6·8-s + 49.8·9-s + 31.6i·10-s − 138. i·11-s − 91.5·12-s − 27.6·13-s − 29.2i·14-s − 127. i·15-s + 64.0·16-s + 79.2i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.27·3-s + 0.500·4-s + 0.447i·5-s − 0.898·6-s − 0.211i·7-s + 0.353·8-s + 0.615·9-s + 0.316i·10-s − 1.14i·11-s − 0.635·12-s − 0.163·13-s − 0.149i·14-s − 0.568i·15-s + 0.250·16-s + 0.274i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.932 - 0.362i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.932 - 0.362i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.820535546\)
\(L(\frac12)\) \(\approx\) \(1.820535546\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82T \)
5 \( 1 - 11.1iT \)
23 \( 1 + (-191. - 493. i)T \)
good3 \( 1 + 11.4T + 81T^{2} \)
7 \( 1 + 10.3iT - 2.40e3T^{2} \)
11 \( 1 + 138. iT - 1.46e4T^{2} \)
13 \( 1 + 27.6T + 2.85e4T^{2} \)
17 \( 1 - 79.2iT - 8.35e4T^{2} \)
19 \( 1 - 362. iT - 1.30e5T^{2} \)
29 \( 1 - 1.33e3T + 7.07e5T^{2} \)
31 \( 1 - 1.75e3T + 9.23e5T^{2} \)
37 \( 1 + 470. iT - 1.87e6T^{2} \)
41 \( 1 + 1.57e3T + 2.82e6T^{2} \)
43 \( 1 + 1.69e3iT - 3.41e6T^{2} \)
47 \( 1 - 4.01e3T + 4.87e6T^{2} \)
53 \( 1 - 5.03e3iT - 7.89e6T^{2} \)
59 \( 1 + 947.T + 1.21e7T^{2} \)
61 \( 1 - 2.73e3iT - 1.38e7T^{2} \)
67 \( 1 - 3.85e3iT - 2.01e7T^{2} \)
71 \( 1 - 788.T + 2.54e7T^{2} \)
73 \( 1 + 4.54e3T + 2.83e7T^{2} \)
79 \( 1 + 5.55e3iT - 3.89e7T^{2} \)
83 \( 1 - 6.10e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.39e4iT - 6.27e7T^{2} \)
97 \( 1 - 9.36e3iT - 8.85e7T^{2} \)
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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

−11.81941930472597509014521360521, −10.73298537535454425686587006848, −10.26988287871671823279845899974, −8.515150988555736214706633634501, −7.20484943228082414101829334858, −6.14704282908303773760811011299, −5.59510186037232570919078286647, −4.29757897624092521638548967739, −2.96610024683745503383794340531, −0.954670151745896389006286122863, 0.78483134890989625469952713396, 2.58297235256497838224479514398, 4.61579870122560186732980276878, 4.95542885073154834928350008716, 6.28084683027367008490623529932, 6.99168416707689066162316319148, 8.467688654662413126038525867147, 9.867354180347194309091943452403, 10.73608734082919618028472774410, 11.86981656034946289687857970966

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