Properties

Label 2-230-23.22-c4-0-11
Degree $2$
Conductor $230$
Sign $0.396 + 0.918i$
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 − 14.3·3-s + 8.00·4-s − 11.1i·5-s + 40.5·6-s − 16.7i·7-s − 22.6·8-s + 124.·9-s + 31.6i·10-s + 34.6i·11-s − 114.·12-s − 97.0·13-s + 47.3i·14-s + 160. i·15-s + 64.0·16-s + 198. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.59·3-s + 0.500·4-s − 0.447i·5-s + 1.12·6-s − 0.341i·7-s − 0.353·8-s + 1.53·9-s + 0.316i·10-s + 0.286i·11-s − 0.795·12-s − 0.574·13-s + 0.241i·14-s + 0.711i·15-s + 0.250·16-s + 0.686i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.396 + 0.918i$
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: \((2,\ 230,\ (\ :2),\ 0.396 + 0.918i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4657870437\)
\(L(\frac12)\) \(\approx\) \(0.4657870437\)
\(L(3)\) 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 \( 1 + (485. - 209. i)T \)
good3 \( 1 + 14.3T + 81T^{2} \)
7 \( 1 + 16.7iT - 2.40e3T^{2} \)
11 \( 1 - 34.6iT - 1.46e4T^{2} \)
13 \( 1 + 97.0T + 2.85e4T^{2} \)
17 \( 1 - 198. iT - 8.35e4T^{2} \)
19 \( 1 - 451. iT - 1.30e5T^{2} \)
29 \( 1 - 431.T + 7.07e5T^{2} \)
31 \( 1 - 67.9T + 9.23e5T^{2} \)
37 \( 1 + 385. iT - 1.87e6T^{2} \)
41 \( 1 - 542.T + 2.82e6T^{2} \)
43 \( 1 + 1.23e3iT - 3.41e6T^{2} \)
47 \( 1 + 112.T + 4.87e6T^{2} \)
53 \( 1 + 2.77e3iT - 7.89e6T^{2} \)
59 \( 1 - 2.87e3T + 1.21e7T^{2} \)
61 \( 1 + 5.98e3iT - 1.38e7T^{2} \)
67 \( 1 + 352. iT - 2.01e7T^{2} \)
71 \( 1 + 5.06e3T + 2.54e7T^{2} \)
73 \( 1 + 144.T + 2.83e7T^{2} \)
79 \( 1 - 6.20e3iT - 3.89e7T^{2} \)
83 \( 1 - 5.87e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.56e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.27e4iT - 8.85e7T^{2} \)
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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

−11.32582238769392670286429188478, −10.32852232225266488676197953347, −9.818444704330221708561778119041, −8.331357949803382202115072691275, −7.27912783489071291686704337890, −6.21156806665110522691693895350, −5.34825051365554799869713466740, −4.06889482531240177526028000537, −1.69956523188949675293404056744, −0.38416353038608585848712599947, 0.75521546841694965623877861951, 2.60044825384149277063500824299, 4.62214461983816737733279562805, 5.75394096544211895681225913933, 6.62675779981236611723897278967, 7.50396158840494542411977559579, 8.945452879289797456620362664169, 10.05550060794381085441989355962, 10.76970702802378248487489535249, 11.66747408264590445456277065807

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