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Properties

Label 2-230-5.2-c2-0-8
Degree $2$
Conductor $230$
Sign $0.943 - 0.331i$
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 + (0.608 − 0.608i)3-s + 2i·4-s + (4.88 + 1.06i)5-s − 1.21·6-s + (3.48 + 3.48i)7-s + (2 − 2i)8-s + 8.25i·9-s + (−3.81 − 5.95i)10-s − 3.82·11-s + (1.21 + 1.21i)12-s + (−13.1 + 13.1i)13-s − 6.96i·14-s + (3.62 − 2.32i)15-s − 4·16-s + (15.9 + 15.9i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.202 − 0.202i)3-s + 0.5i·4-s + (0.976 + 0.213i)5-s − 0.202·6-s + (0.497 + 0.497i)7-s + (0.250 − 0.250i)8-s + 0.917i·9-s + (−0.381 − 0.595i)10-s − 0.347·11-s + (0.101 + 0.101i)12-s + (−1.00 + 1.00i)13-s − 0.497i·14-s + (0.241 − 0.154i)15-s − 0.250·16-s + (0.940 + 0.940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.943 - 0.331i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.47494 + 0.251715i\)
\(L(\frac12)\) \(\approx\) \(1.47494 + 0.251715i\)
\(L(2)\) 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 \( 1 + (-4.88 - 1.06i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good3 \( 1 + (-0.608 + 0.608i)T - 9iT^{2} \)
7 \( 1 + (-3.48 - 3.48i)T + 49iT^{2} \)
11 \( 1 + 3.82T + 121T^{2} \)
13 \( 1 + (13.1 - 13.1i)T - 169iT^{2} \)
17 \( 1 + (-15.9 - 15.9i)T + 289iT^{2} \)
19 \( 1 + 6.16iT - 361T^{2} \)
29 \( 1 + 32.8iT - 841T^{2} \)
31 \( 1 - 27.0T + 961T^{2} \)
37 \( 1 + (20.0 + 20.0i)T + 1.36e3iT^{2} \)
41 \( 1 - 17.5T + 1.68e3T^{2} \)
43 \( 1 + (-26.6 + 26.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-33.3 - 33.3i)T + 2.20e3iT^{2} \)
53 \( 1 + (42.6 - 42.6i)T - 2.80e3iT^{2} \)
59 \( 1 + 28.9iT - 3.48e3T^{2} \)
61 \( 1 + 3.95T + 3.72e3T^{2} \)
67 \( 1 + (69.4 + 69.4i)T + 4.48e3iT^{2} \)
71 \( 1 - 65.0T + 5.04e3T^{2} \)
73 \( 1 + (52.2 - 52.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 62.2iT - 6.24e3T^{2} \)
83 \( 1 + (95.2 - 95.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 131. iT - 7.92e3T^{2} \)
97 \( 1 + (-20.1 - 20.1i)T + 9.40e3iT^{2} \)
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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

−12.00208853593105805448551776641, −10.90081615374503758320820570627, −10.11443380923685377065769019750, −9.221291477178556009453652147602, −8.190742875900924888023294433736, −7.24242787436539176015588816861, −5.83057203765564601436525065201, −4.63955395268705536643847781966, −2.63053247636160745050417495089, −1.80457608641242588420659668517, 1.01054587406549132263765999502, 2.92108203924026409561888730365, 4.83697059420439752616142325743, 5.72426158544205698406343011470, 7.00723049946409093219669846838, 7.955050288471832313941991465758, 9.096587068532799714063800986531, 9.919538820166393384063676971966, 10.50098447393298045093831757055, 11.96957200857967595722089165544

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