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Properties

Label 2-230-5.4-c3-0-23
Degree $2$
Conductor $230$
Sign $-0.567 + 0.823i$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 2.36i·3-s − 4·4-s + (−6.34 + 9.20i)5-s + 4.72·6-s + 7.27i·7-s − 8i·8-s + 21.4·9-s + (−18.4 − 12.6i)10-s − 38.4·11-s + 9.45i·12-s + 9.13i·13-s − 14.5·14-s + (21.7 + 14.9i)15-s + 16·16-s − 75.3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.454i·3-s − 0.5·4-s + (−0.567 + 0.823i)5-s + 0.321·6-s + 0.393i·7-s − 0.353i·8-s + 0.793·9-s + (−0.582 − 0.401i)10-s − 1.05·11-s + 0.227i·12-s + 0.194i·13-s − 0.277·14-s + (0.374 + 0.258i)15-s + 0.250·16-s − 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.567 + 0.823i$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ -0.567 + 0.823i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0112568 - 0.0214259i\)
\(L(\frac12)\) \(\approx\) \(0.0112568 - 0.0214259i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (6.34 - 9.20i)T \)
23 \( 1 + 23iT \)
good3 \( 1 + 2.36iT - 27T^{2} \)
7 \( 1 - 7.27iT - 343T^{2} \)
11 \( 1 + 38.4T + 1.33e3T^{2} \)
13 \( 1 - 9.13iT - 2.19e3T^{2} \)
17 \( 1 + 75.3iT - 4.91e3T^{2} \)
19 \( 1 + 94.6T + 6.85e3T^{2} \)
29 \( 1 + 195.T + 2.43e4T^{2} \)
31 \( 1 + 213.T + 2.97e4T^{2} \)
37 \( 1 + 277. iT - 5.06e4T^{2} \)
41 \( 1 + 190.T + 6.89e4T^{2} \)
43 \( 1 - 560. iT - 7.95e4T^{2} \)
47 \( 1 + 151. iT - 1.03e5T^{2} \)
53 \( 1 + 613. iT - 1.48e5T^{2} \)
59 \( 1 - 470.T + 2.05e5T^{2} \)
61 \( 1 + 859.T + 2.26e5T^{2} \)
67 \( 1 - 299. iT - 3.00e5T^{2} \)
71 \( 1 - 1.11e3T + 3.57e5T^{2} \)
73 \( 1 - 416. iT - 3.89e5T^{2} \)
79 \( 1 + 993.T + 4.93e5T^{2} \)
83 \( 1 - 870. iT - 5.71e5T^{2} \)
89 \( 1 - 613.T + 7.04e5T^{2} \)
97 \( 1 - 489. iT - 9.12e5T^{2} \)
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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

−11.41436208362772053927945308523, −10.47127214743669587621031471414, −9.365455452693571506349078073487, −8.087003520446524772484130839823, −7.34370954172581514910092694877, −6.58498583285462086163639985979, −5.28611564514115891434684493224, −3.94565776378535071589361113164, −2.35024047293773266478646282572, −0.009668422986229320458619240956, 1.71100093458344807556220007236, 3.63547162086587471995957655029, 4.43742887406748722469667133467, 5.52412050377236507637384495075, 7.34909410172777800123058713432, 8.321113692275161766013769188633, 9.281876878673962255581214349565, 10.43436891088159139373581038450, 10.85067611443690884030278486364, 12.22670496827424704605833303345

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