↑

Properties

Label 2-230-5.3-c2-0-15
Degree $2$
Conductor $230$
Sign $-0.804 + 0.593i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−2.63 − 2.63i)3-s − 2i·4-s + (1.86 − 4.64i)5-s + 5.27·6-s + (5.81 − 5.81i)7-s + (2 + 2i)8-s + 4.93i·9-s + (2.77 + 6.50i)10-s − 12.5·11-s + (−5.27 + 5.27i)12-s + (9.20 + 9.20i)13-s + 11.6i·14-s + (−17.1 + 7.33i)15-s − 4·16-s + (−1.16 + 1.16i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.879 − 0.879i)3-s − 0.5i·4-s + (0.372 − 0.928i)5-s + 0.879·6-s + (0.830 − 0.830i)7-s + (0.250 + 0.250i)8-s + 0.548i·9-s + (0.277 + 0.650i)10-s − 1.14·11-s + (−0.439 + 0.439i)12-s + (0.707 + 0.707i)13-s + 0.830i·14-s + (−1.14 + 0.488i)15-s − 0.250·16-s + (−0.0685 + 0.0685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.804 + 0.593i$
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: \((2,\ 230,\ (\ :1),\ -0.804 + 0.593i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.223134 - 0.678454i\)
\(L(\frac12)\) \(\approx\) \(0.223134 - 0.678454i\)
\(L(2)\) 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 \( 1 + (-1.86 + 4.64i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good3 \( 1 + (2.63 + 2.63i)T + 9iT^{2} \)
7 \( 1 + (-5.81 + 5.81i)T - 49iT^{2} \)
11 \( 1 + 12.5T + 121T^{2} \)
13 \( 1 + (-9.20 - 9.20i)T + 169iT^{2} \)
17 \( 1 + (1.16 - 1.16i)T - 289iT^{2} \)
19 \( 1 + 28.2iT - 361T^{2} \)
29 \( 1 + 4.28iT - 841T^{2} \)
31 \( 1 + 33.8T + 961T^{2} \)
37 \( 1 + (3.35 - 3.35i)T - 1.36e3iT^{2} \)
41 \( 1 + 30.6T + 1.68e3T^{2} \)
43 \( 1 + (-39.8 - 39.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (20.8 - 20.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (11.2 + 11.2i)T + 2.80e3iT^{2} \)
59 \( 1 + 42.8iT - 3.48e3T^{2} \)
61 \( 1 - 25.6T + 3.72e3T^{2} \)
67 \( 1 + (-68.3 + 68.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 33.1T + 5.04e3T^{2} \)
73 \( 1 + (-53.4 - 53.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 43.7iT - 6.24e3T^{2} \)
83 \( 1 + (72.3 + 72.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 158. iT - 7.92e3T^{2} \)
97 \( 1 + (-33.4 + 33.4i)T - 9.40e3iT^{2} \)
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

−11.37420700574410126912954395715, −10.87445292892686329852882833739, −9.520130234121843888848586699580, −8.433927989595803195973625255191, −7.53007905168954407828552725156, −6.58888718373938901225247678187, −5.48621420825398070864717687797, −4.58711784246854487245989147446, −1.71107885784174436209770684247, −0.50873244932290929484528463898, 2.13212733718857749743180218063, 3.61601000863730521934054516200, 5.26578883639149132224525725907, 5.87409033344590350609296582505, 7.58502883541825312978612075562, 8.544089313199134509094836893617, 9.916970773621181165329869961874, 10.52737589392374017617721288957, 11.09687504669361572194923114874, 11.93725081767606671466760573689

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