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Properties

Label 2-230-5.3-c4-0-1
Degree $2$
Conductor $230$
Sign $-0.343 + 0.939i$
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 + (6.67 + 6.67i)3-s − 8i·4-s + (5.03 + 24.4i)5-s − 26.6·6-s + (−43.7 + 43.7i)7-s + (16 + 16i)8-s + 8.02i·9-s + (−59.0 − 38.8i)10-s − 158.·11-s + (53.3 − 53.3i)12-s + (−13.0 − 13.0i)13-s − 175. i·14-s + (−129. + 196. i)15-s − 64·16-s + (108. − 108. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.741 + 0.741i)3-s − 0.5i·4-s + (0.201 + 0.979i)5-s − 0.741·6-s + (−0.892 + 0.892i)7-s + (0.250 + 0.250i)8-s + 0.0990i·9-s + (−0.590 − 0.388i)10-s − 1.30·11-s + (0.370 − 0.370i)12-s + (−0.0773 − 0.0773i)13-s − 0.892i·14-s + (−0.576 + 0.875i)15-s − 0.250·16-s + (0.375 − 0.375i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.343 + 0.939i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.343 + 0.939i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4793619908\)
\(L(\frac12)\) \(\approx\) \(0.4793619908\)
\(L(3)\) 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 \( 1 + (-5.03 - 24.4i)T \)
23 \( 1 + (77.9 + 77.9i)T \)
good3 \( 1 + (-6.67 - 6.67i)T + 81iT^{2} \)
7 \( 1 + (43.7 - 43.7i)T - 2.40e3iT^{2} \)
11 \( 1 + 158.T + 1.46e4T^{2} \)
13 \( 1 + (13.0 + 13.0i)T + 2.85e4iT^{2} \)
17 \( 1 + (-108. + 108. i)T - 8.35e4iT^{2} \)
19 \( 1 + 274. iT - 1.30e5T^{2} \)
29 \( 1 - 869. iT - 7.07e5T^{2} \)
31 \( 1 - 77.4T + 9.23e5T^{2} \)
37 \( 1 + (1.18e3 - 1.18e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 428.T + 2.82e6T^{2} \)
43 \( 1 + (-601. - 601. i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.75e3 + 1.75e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (52.8 + 52.8i)T + 7.89e6iT^{2} \)
59 \( 1 + 3.67e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.01e3T + 1.38e7T^{2} \)
67 \( 1 + (-4.08e3 + 4.08e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 9.04e3T + 2.54e7T^{2} \)
73 \( 1 + (4.87e3 + 4.87e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 1.00e4iT - 3.89e7T^{2} \)
83 \( 1 + (2.99e3 + 2.99e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 784. iT - 6.27e7T^{2} \)
97 \( 1 + (1.07e4 - 1.07e4i)T - 8.85e7iT^{2} \)
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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

−12.18551325637775240517156721143, −10.78796610115076240321294134093, −10.04668702765608399810889701361, −9.343440983840233033774349262310, −8.458027657391581937785567828084, −7.26740316447811145296983318446, −6.24400810069874039293883021241, −5.11026164452639600363849692192, −3.26616333221163108952381286678, −2.55272159199291704516394824483, 0.16566710622053209292029436759, 1.51079428331813919271835042115, 2.76389472444940420415396806128, 4.12171293220227689807312070329, 5.71169264812732232473421167166, 7.30058471521235123796534728091, 7.938853898234510586098321704873, 8.829320623459969338256953312017, 9.950475234434193487493096860145, 10.55785212967920150668472757719

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