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Properties

Label 2-230-5.2-c4-0-16
Degree $2$
Conductor $230$
Sign $-0.0318 - 0.999i$
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 + (7.12 − 7.12i)3-s + 8i·4-s + (−12.4 + 21.6i)5-s + 28.4·6-s + (−34.8 − 34.8i)7-s + (−16 + 16i)8-s − 20.5i·9-s + (−68.2 + 18.4i)10-s + 168.·11-s + (56.9 + 56.9i)12-s + (−92.5 + 92.5i)13-s − 139. i·14-s + (65.6 + 243. i)15-s − 64·16-s + (353. + 353. i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.791 − 0.791i)3-s + 0.5i·4-s + (−0.498 + 0.866i)5-s + 0.791·6-s + (−0.711 − 0.711i)7-s + (−0.250 + 0.250i)8-s − 0.253i·9-s + (−0.682 + 0.184i)10-s + 1.39·11-s + (0.395 + 0.395i)12-s + (−0.547 + 0.547i)13-s − 0.711i·14-s + (0.291 + 1.08i)15-s − 0.250·16-s + (1.22 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0318 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0318 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.0318 - 0.999i$
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: \((2,\ 230,\ (\ :2),\ -0.0318 - 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.590277504\)
\(L(\frac12)\) \(\approx\) \(2.590277504\)
\(L(3)\) 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 \( 1 + (12.4 - 21.6i)T \)
23 \( 1 + (77.9 - 77.9i)T \)
good3 \( 1 + (-7.12 + 7.12i)T - 81iT^{2} \)
7 \( 1 + (34.8 + 34.8i)T + 2.40e3iT^{2} \)
11 \( 1 - 168.T + 1.46e4T^{2} \)
13 \( 1 + (92.5 - 92.5i)T - 2.85e4iT^{2} \)
17 \( 1 + (-353. - 353. i)T + 8.35e4iT^{2} \)
19 \( 1 - 470. iT - 1.30e5T^{2} \)
29 \( 1 - 1.12e3iT - 7.07e5T^{2} \)
31 \( 1 + 1.17e3T + 9.23e5T^{2} \)
37 \( 1 + (707. + 707. i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.86e3T + 2.82e6T^{2} \)
43 \( 1 + (604. - 604. i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.41e3 - 1.41e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-2.23e3 + 2.23e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 1.59e3iT - 1.21e7T^{2} \)
61 \( 1 - 198.T + 1.38e7T^{2} \)
67 \( 1 + (3.61e3 + 3.61e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 4.49e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.31e3 + 3.31e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 8.58e3iT - 3.89e7T^{2} \)
83 \( 1 + (7.43e3 - 7.43e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 161. iT - 6.27e7T^{2} \)
97 \( 1 + (4.58e3 + 4.58e3i)T + 8.85e7iT^{2} \)
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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

−12.24176577899485538825105837452, −10.89835829576179275337087540740, −9.780595744830352616051851039521, −8.504249805076365083149139262645, −7.46014518826137689834203427952, −6.99194726003890006645920918440, −5.97569624921437162649911192816, −3.92319621495755151499595405502, −3.36036281132332240287789958255, −1.66575105969261071986155678498, 0.69011085081962313712768538223, 2.69983256380724761658778484520, 3.66343311135850825757367860811, 4.64814135367195506209441551101, 5.80240965893820232734978410992, 7.34667896688035874307071120045, 8.917102181793364927594399853385, 9.244302967407120492367220952762, 10.07360281829050018578809135227, 11.70943221057999663151872690255

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