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Properties

Label 2-230-5.2-c4-0-19
Degree $2$
Conductor $230$
Sign $0.602 + 0.798i$
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 − 2i)2-s + (−11.8 + 11.8i)3-s + 8i·4-s + (−2.32 − 24.8i)5-s + 47.2·6-s + (62.0 + 62.0i)7-s + (16 − 16i)8-s − 197. i·9-s + (−45.1 + 54.4i)10-s − 179.·11-s + (−94.4 − 94.4i)12-s + (−141. + 141. i)13-s − 248. i·14-s + (321. + 266. i)15-s − 64·16-s + (−88.6 − 88.6i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−1.31 + 1.31i)3-s + 0.5i·4-s + (−0.0929 − 0.995i)5-s + 1.31·6-s + (1.26 + 1.26i)7-s + (0.250 − 0.250i)8-s − 2.44i·9-s + (−0.451 + 0.544i)10-s − 1.48·11-s + (−0.655 − 0.655i)12-s + (−0.838 + 0.838i)13-s − 1.26i·14-s + (1.42 + 1.18i)15-s − 0.250·16-s + (−0.306 − 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.602 + 0.798i$
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: \((2,\ 230,\ (\ :2),\ 0.602 + 0.798i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4627360352\)
\(L(\frac12)\) \(\approx\) \(0.4627360352\)
\(L(3)\) 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 \( 1 + (2.32 + 24.8i)T \)
23 \( 1 + (-77.9 + 77.9i)T \)
good3 \( 1 + (11.8 - 11.8i)T - 81iT^{2} \)
7 \( 1 + (-62.0 - 62.0i)T + 2.40e3iT^{2} \)
11 \( 1 + 179.T + 1.46e4T^{2} \)
13 \( 1 + (141. - 141. i)T - 2.85e4iT^{2} \)
17 \( 1 + (88.6 + 88.6i)T + 8.35e4iT^{2} \)
19 \( 1 - 78.5iT - 1.30e5T^{2} \)
29 \( 1 + 208. iT - 7.07e5T^{2} \)
31 \( 1 + 789.T + 9.23e5T^{2} \)
37 \( 1 + (1.35e3 + 1.35e3i)T + 1.87e6iT^{2} \)
41 \( 1 - 2.97e3T + 2.82e6T^{2} \)
43 \( 1 + (-2.27e3 + 2.27e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.20e3 - 1.20e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (622. - 622. i)T - 7.89e6iT^{2} \)
59 \( 1 + 3.40e3iT - 1.21e7T^{2} \)
61 \( 1 + 456.T + 1.38e7T^{2} \)
67 \( 1 + (-444. - 444. i)T + 2.01e7iT^{2} \)
71 \( 1 - 805.T + 2.54e7T^{2} \)
73 \( 1 + (-4.30e3 + 4.30e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 6.62e3iT - 3.89e7T^{2} \)
83 \( 1 + (-7.45e3 + 7.45e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 2.82e3iT - 6.27e7T^{2} \)
97 \( 1 + (-3.36e3 - 3.36e3i)T + 8.85e7iT^{2} \)
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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

−11.24626791773957888894067380079, −10.66382013871635137141635473160, −9.412347150723952613589490004701, −8.935413420881581077194528480210, −7.67808108508401915693817268238, −5.64986274674579877053700860490, −5.08337045676515436218372933004, −4.28386233932485787146436143200, −2.18948340288159891828335087042, −0.30239200794824458724698262251, 0.842520812845015662572944315419, 2.27368212290667277503010578986, 4.82523421386699187416968062967, 5.70309564842393533331651875679, 6.93837466790915295780020391160, 7.61446031253249459678535687070, 7.915803233265305418797020894550, 10.33125544397247067696188789521, 10.75784404541111254236805319941, 11.34126081655856398100420670897

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