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Properties

Label 2-230-5.2-c4-0-8
Degree $2$
Conductor $230$
Sign $0.979 - 0.202i$
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 + (−2.34 + 2.34i)3-s + 8i·4-s + (−23.4 − 8.55i)5-s + 9.38·6-s + (−59.8 − 59.8i)7-s + (16 − 16i)8-s + 69.9i·9-s + (29.8 + 64.0i)10-s − 73.2·11-s + (−18.7 − 18.7i)12-s + (−43.3 + 43.3i)13-s + 239. i·14-s + (75.1 − 35.0i)15-s − 64·16-s + (−284. − 284. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.260 + 0.260i)3-s + 0.5i·4-s + (−0.939 − 0.342i)5-s + 0.260·6-s + (−1.22 − 1.22i)7-s + (0.250 − 0.250i)8-s + 0.864i·9-s + (0.298 + 0.640i)10-s − 0.604·11-s + (−0.130 − 0.130i)12-s + (−0.256 + 0.256i)13-s + 1.22i·14-s + (0.334 − 0.155i)15-s − 0.250·16-s + (−0.985 − 0.985i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.979 - 0.202i$
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: \((2,\ 230,\ (\ :2),\ 0.979 - 0.202i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4028177501\)
\(L(\frac12)\) \(\approx\) \(0.4028177501\)
\(L(3)\) 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 \( 1 + (23.4 + 8.55i)T \)
23 \( 1 + (-77.9 + 77.9i)T \)
good3 \( 1 + (2.34 - 2.34i)T - 81iT^{2} \)
7 \( 1 + (59.8 + 59.8i)T + 2.40e3iT^{2} \)
11 \( 1 + 73.2T + 1.46e4T^{2} \)
13 \( 1 + (43.3 - 43.3i)T - 2.85e4iT^{2} \)
17 \( 1 + (284. + 284. i)T + 8.35e4iT^{2} \)
19 \( 1 + 280. iT - 1.30e5T^{2} \)
29 \( 1 - 1.45e3iT - 7.07e5T^{2} \)
31 \( 1 + 425.T + 9.23e5T^{2} \)
37 \( 1 + (131. + 131. i)T + 1.87e6iT^{2} \)
41 \( 1 - 2.38e3T + 2.82e6T^{2} \)
43 \( 1 + (-842. + 842. i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.59e3 - 1.59e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-498. + 498. i)T - 7.89e6iT^{2} \)
59 \( 1 + 5.05e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.10e3T + 1.38e7T^{2} \)
67 \( 1 + (-3.22e3 - 3.22e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 1.16e3T + 2.54e7T^{2} \)
73 \( 1 + (1.37e3 - 1.37e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 103. iT - 3.89e7T^{2} \)
83 \( 1 + (1.87e3 - 1.87e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 7.52e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.10e4 + 1.10e4i)T + 8.85e7iT^{2} \)
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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

−11.12953446696101824821916791124, −10.84901831103287483394285225253, −9.732108318359000773819814179067, −8.805095927547678921504809203360, −7.47532042162826741809764169270, −6.97056401717027376458720973590, −4.96990613820904460524923460347, −4.01539696312616783726939982739, −2.74353477482966327238551039364, −0.60419781493032208907120192347, 0.30838536385688504093082537780, 2.55914981601124302072251069893, 3.93654224781295101388462467162, 5.78894976834145811501825270737, 6.38105146866641338352564117540, 7.48916245076406277346514386320, 8.538033815017734260037551970971, 9.411968687924346524141857696157, 10.43735876294093648209887293212, 11.60039626713797178380158972145

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