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Properties

Label 2-230-5.3-c2-0-17
Degree $2$
Conductor $230$
Sign $-0.946 + 0.323i$
Analytic cond. $6.26704$
Root an. cond. $2.50340$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (−1.36 − 1.36i)3-s − 2i·4-s + (−3.17 + 3.86i)5-s − 2.72·6-s + (7.21 − 7.21i)7-s + (−2 − 2i)8-s − 5.29i·9-s + (0.691 + 7.03i)10-s − 14.8·11-s + (−2.72 + 2.72i)12-s + (−4.91 − 4.91i)13-s − 14.4i·14-s + (9.57 − 0.941i)15-s − 4·16-s + (−17.8 + 17.8i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.453 − 0.453i)3-s − 0.5i·4-s + (−0.634 + 0.772i)5-s − 0.453·6-s + (1.03 − 1.03i)7-s + (−0.250 − 0.250i)8-s − 0.588i·9-s + (0.0691 + 0.703i)10-s − 1.34·11-s + (−0.226 + 0.226i)12-s + (−0.377 − 0.377i)13-s − 1.03i·14-s + (0.638 − 0.0627i)15-s − 0.250·16-s + (−1.04 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.946 + 0.323i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ -0.946 + 0.323i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.181275 - 1.08924i\)
\(L(\frac12)\) \(\approx\) \(0.181275 - 1.08924i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (3.17 - 3.86i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good3 \( 1 + (1.36 + 1.36i)T + 9iT^{2} \)
7 \( 1 + (-7.21 + 7.21i)T - 49iT^{2} \)
11 \( 1 + 14.8T + 121T^{2} \)
13 \( 1 + (4.91 + 4.91i)T + 169iT^{2} \)
17 \( 1 + (17.8 - 17.8i)T - 289iT^{2} \)
19 \( 1 + 29.6iT - 361T^{2} \)
29 \( 1 - 11.0iT - 841T^{2} \)
31 \( 1 - 42.3T + 961T^{2} \)
37 \( 1 + (-14.7 + 14.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 23.9T + 1.68e3T^{2} \)
43 \( 1 + (-0.419 - 0.419i)T + 1.84e3iT^{2} \)
47 \( 1 + (-53.0 + 53.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (58.3 + 58.3i)T + 2.80e3iT^{2} \)
59 \( 1 + 13.7iT - 3.48e3T^{2} \)
61 \( 1 - 39.0T + 3.72e3T^{2} \)
67 \( 1 + (16.9 - 16.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 103.T + 5.04e3T^{2} \)
73 \( 1 + (-41.5 - 41.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 155. iT - 6.24e3T^{2} \)
83 \( 1 + (-12.7 - 12.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 141. iT - 7.92e3T^{2} \)
97 \( 1 + (29.4 - 29.4i)T - 9.40e3iT^{2} \)
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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

−11.28331719494161076959792003112, −11.00101895203364225937239134179, −10.11461792433378073739363093932, −8.361173341113829446140879992087, −7.35056963224485049386170024898, −6.51901985802673595845716900290, −5.03230963565957046736233204548, −3.99798075929189135814454529576, −2.49708523012231786198323589377, −0.51972224347904153347236124230, 2.39554648866499812083991091566, 4.48500638316649469321961188967, 4.97667931117871095104984899602, 5.86863426322771119080410872347, 7.75138950313583032301988395001, 8.120114627370094902119858347466, 9.338387415303841278853875731044, 10.74408780500122409375658643815, 11.66544766923847093109492523133, 12.24700116513579883550273364562

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