↑

Properties

Label 2-230-5.2-c4-0-41
Degree $2$
Conductor $230$
Sign $-0.792 - 0.609i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s + (12.2 − 12.2i)3-s + 8i·4-s + (8.85 + 23.3i)5-s − 48.9·6-s + (−54.9 − 54.9i)7-s + (16 − 16i)8-s − 217. i·9-s + (29.0 − 64.4i)10-s − 61.2·11-s + (97.8 + 97.8i)12-s + (−227. + 227. i)13-s + 219. i·14-s + (394. + 177. i)15-s − 64·16-s + (133. + 133. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (1.35 − 1.35i)3-s + 0.5i·4-s + (0.354 + 0.935i)5-s − 1.35·6-s + (−1.12 − 1.12i)7-s + (0.250 − 0.250i)8-s − 2.69i·9-s + (0.290 − 0.644i)10-s − 0.506·11-s + (0.679 + 0.679i)12-s + (−1.34 + 1.34i)13-s + 1.12i·14-s + (1.75 + 0.789i)15-s − 0.250·16-s + (0.461 + 0.461i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.792 - 0.609i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.792 - 0.609i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9524685867\)
\(L(\frac12)\) \(\approx\) \(0.9524685867\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
5 \( 1 + (-8.85 - 23.3i)T \)
23 \( 1 + (-77.9 + 77.9i)T \)
good3 \( 1 + (-12.2 + 12.2i)T - 81iT^{2} \)
7 \( 1 + (54.9 + 54.9i)T + 2.40e3iT^{2} \)
11 \( 1 + 61.2T + 1.46e4T^{2} \)
13 \( 1 + (227. - 227. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-133. - 133. i)T + 8.35e4iT^{2} \)
19 \( 1 + 424. iT - 1.30e5T^{2} \)
29 \( 1 + 509. iT - 7.07e5T^{2} \)
31 \( 1 + 1.32e3T + 9.23e5T^{2} \)
37 \( 1 + (560. + 560. i)T + 1.87e6iT^{2} \)
41 \( 1 - 957.T + 2.82e6T^{2} \)
43 \( 1 + (-269. + 269. i)T - 3.41e6iT^{2} \)
47 \( 1 + (843. + 843. i)T + 4.87e6iT^{2} \)
53 \( 1 + (979. - 979. i)T - 7.89e6iT^{2} \)
59 \( 1 - 832. iT - 1.21e7T^{2} \)
61 \( 1 + 148.T + 1.38e7T^{2} \)
67 \( 1 + (3.37e3 + 3.37e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 8.35e3T + 2.54e7T^{2} \)
73 \( 1 + (-4.32e3 + 4.32e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 4.40e3iT - 3.89e7T^{2} \)
83 \( 1 + (-7.62e3 + 7.62e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.05e4iT - 6.27e7T^{2} \)
97 \( 1 + (2.80e3 + 2.80e3i)T + 8.85e7iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.83324528022440740285162746937, −9.666154872669500842072438705389, −9.246413058149884150345316432813, −7.72059012622144086198757327800, −7.11814956684882237049441360963, −6.56773840041301770614894489527, −3.79523464797039864198332014839, −2.81031576300385068652603972469, −1.95247130186812133633358084750, −0.28572095246217832843796268501, 2.33074182139113891765321816861, 3.35837724673241552645439622894, 5.06507039374335189411212261868, 5.58069594075131045686192649751, 7.67035860165910360771304872452, 8.405492776301962972345892184459, 9.384829221828500832556439342580, 9.696341316782878481471709533777, 10.47938179243092457870555800018, 12.41611346879082528324600190084

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