Properties

Label 2-230-5.3-c4-0-20
Degree $2$
Conductor $230$
Sign $-0.824 + 0.566i$
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 + (−9.43 − 9.43i)3-s − 8i·4-s + (−10.0 + 22.8i)5-s + 37.7·6-s + (−36.9 + 36.9i)7-s + (16 + 16i)8-s + 97.1i·9-s + (−25.5 − 65.9i)10-s + 159.·11-s + (−75.4 + 75.4i)12-s + (3.50 + 3.50i)13-s − 147. i·14-s + (311. − 120. i)15-s − 64·16-s + (307. − 307. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−1.04 − 1.04i)3-s − 0.5i·4-s + (−0.403 + 0.915i)5-s + 1.04·6-s + (−0.753 + 0.753i)7-s + (0.250 + 0.250i)8-s + 1.19i·9-s + (−0.255 − 0.659i)10-s + 1.32·11-s + (−0.524 + 0.524i)12-s + (0.0207 + 0.0207i)13-s − 0.753i·14-s + (1.38 − 0.536i)15-s − 0.250·16-s + (1.06 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.566i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.824 + 0.566i$
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: \((2,\ 230,\ (\ :2),\ -0.824 + 0.566i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.03115762672\)
\(L(\frac12)\) \(\approx\) \(0.03115762672\)
\(L(3)\) 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 \( 1 + (10.0 - 22.8i)T \)
23 \( 1 + (77.9 + 77.9i)T \)
good3 \( 1 + (9.43 + 9.43i)T + 81iT^{2} \)
7 \( 1 + (36.9 - 36.9i)T - 2.40e3iT^{2} \)
11 \( 1 - 159.T + 1.46e4T^{2} \)
13 \( 1 + (-3.50 - 3.50i)T + 2.85e4iT^{2} \)
17 \( 1 + (-307. + 307. i)T - 8.35e4iT^{2} \)
19 \( 1 - 390. iT - 1.30e5T^{2} \)
29 \( 1 - 1.17e3iT - 7.07e5T^{2} \)
31 \( 1 + 661.T + 9.23e5T^{2} \)
37 \( 1 + (529. - 529. i)T - 1.87e6iT^{2} \)
41 \( 1 - 644.T + 2.82e6T^{2} \)
43 \( 1 + (1.34e3 + 1.34e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (1.92e3 - 1.92e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (2.34e3 + 2.34e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 2.23e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.18e3T + 1.38e7T^{2} \)
67 \( 1 + (-2.11e3 + 2.11e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 3.84e3T + 2.54e7T^{2} \)
73 \( 1 + (1.12e3 + 1.12e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.14e4iT - 3.89e7T^{2} \)
83 \( 1 + (4.76e3 + 4.76e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 4.13e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.64e3 - 1.64e3i)T - 8.85e7iT^{2} \)
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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

−11.39865411741658368633134104199, −10.19060856695406417271347576999, −9.186274455180375749239314603915, −7.81831768140529590379888487409, −6.87720061306772748233974224427, −6.36536571049211302964352402143, −5.43028365387157608770737385144, −3.34838138399645728584276123859, −1.52547267288975316871068174194, −0.01769719080864776671105877138, 1.10942401992630942847324584272, 3.71122937495850295927863559166, 4.25566900274855954271958751705, 5.61371006251007816740389820416, 6.83228649622760675826464373512, 8.257168393985568709246943451988, 9.469722402837137934527171860055, 9.898642846566086985110031697760, 10.99937573917886018696940120937, 11.69618190497945075266058143951

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