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Properties

Label 2-230-5.3-c4-0-4
Degree $2$
Conductor $230$
Sign $0.983 + 0.181i$
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 + (−11.2 − 11.2i)3-s − 8i·4-s + (−23.2 + 9.07i)5-s − 45.0·6-s + (2.86 − 2.86i)7-s + (−16 − 16i)8-s + 172. i·9-s + (−28.4 + 64.7i)10-s − 25.5·11-s + (−90.0 + 90.0i)12-s + (−91.5 − 91.5i)13-s − 11.4i·14-s + (364. + 160. i)15-s − 64·16-s + (−38.0 + 38.0i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−1.25 − 1.25i)3-s − 0.5i·4-s + (−0.931 + 0.362i)5-s − 1.25·6-s + (0.0584 − 0.0584i)7-s + (−0.250 − 0.250i)8-s + 2.13i·9-s + (−0.284 + 0.647i)10-s − 0.211·11-s + (−0.625 + 0.625i)12-s + (−0.541 − 0.541i)13-s − 0.0584i·14-s + (1.62 + 0.711i)15-s − 0.250·16-s + (−0.131 + 0.131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.983 + 0.181i$
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: \((2,\ 230,\ (\ :2),\ 0.983 + 0.181i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6463850586\)
\(L(\frac12)\) \(\approx\) \(0.6463850586\)
\(L(3)\) 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 \( 1 + (23.2 - 9.07i)T \)
23 \( 1 + (77.9 + 77.9i)T \)
good3 \( 1 + (11.2 + 11.2i)T + 81iT^{2} \)
7 \( 1 + (-2.86 + 2.86i)T - 2.40e3iT^{2} \)
11 \( 1 + 25.5T + 1.46e4T^{2} \)
13 \( 1 + (91.5 + 91.5i)T + 2.85e4iT^{2} \)
17 \( 1 + (38.0 - 38.0i)T - 8.35e4iT^{2} \)
19 \( 1 - 538. iT - 1.30e5T^{2} \)
29 \( 1 + 830. iT - 7.07e5T^{2} \)
31 \( 1 - 214.T + 9.23e5T^{2} \)
37 \( 1 + (-1.38e3 + 1.38e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 366.T + 2.82e6T^{2} \)
43 \( 1 + (1.21e3 + 1.21e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-183. + 183. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-74.7 - 74.7i)T + 7.89e6iT^{2} \)
59 \( 1 - 1.52e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.57e3T + 1.38e7T^{2} \)
67 \( 1 + (5.85e3 - 5.85e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 1.65e3T + 2.54e7T^{2} \)
73 \( 1 + (-5.41e3 - 5.41e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 9.12e3iT - 3.89e7T^{2} \)
83 \( 1 + (-6.84e3 - 6.84e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 4.15e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.55e3 - 1.55e3i)T - 8.85e7iT^{2} \)
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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

−11.72667986405770845126657991096, −10.93583156244475916428172362289, −10.12293536007218599756010244490, −8.095972806118807602412581114186, −7.38995996706189883229130729089, −6.29548629289207667889804876018, −5.41297866268357001457921078028, −4.08822954566693110213426193789, −2.40151302019627670723926642759, −0.876294431036385497592461748722, 0.30030680488077295999352966318, 3.34458345494417433215790247434, 4.72503201387285921067663478141, 4.82183541122012002051694664060, 6.26659441018743880825316090566, 7.29990320550460407713336930448, 8.720526365058833011845426231520, 9.667878030076714475311351119425, 10.91857691237486984380033569995, 11.56307639507115133066533809216

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