Properties

Label 2-230-23.22-c4-0-22
Degree $2$
Conductor $230$
Sign $0.238 + 0.971i$
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  + 2.82·2-s − 1.75·3-s + 8.00·4-s − 11.1i·5-s − 4.96·6-s + 30.3i·7-s + 22.6·8-s − 77.9·9-s − 31.6i·10-s − 137. i·11-s − 14.0·12-s + 219.·13-s + 85.8i·14-s + 19.6i·15-s + 64.0·16-s − 347. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.195·3-s + 0.500·4-s − 0.447i·5-s − 0.137·6-s + 0.619i·7-s + 0.353·8-s − 0.961·9-s − 0.316i·10-s − 1.13i·11-s − 0.0975·12-s + 1.29·13-s + 0.437i·14-s + 0.0872i·15-s + 0.250·16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.238 + 0.971i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.238 + 0.971i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.401205113\)
\(L(\frac12)\) \(\approx\) \(2.401205113\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82T \)
5 \( 1 + 11.1iT \)
23 \( 1 + (-513. + 126. i)T \)
good3 \( 1 + 1.75T + 81T^{2} \)
7 \( 1 - 30.3iT - 2.40e3T^{2} \)
11 \( 1 + 137. iT - 1.46e4T^{2} \)
13 \( 1 - 219.T + 2.85e4T^{2} \)
17 \( 1 + 347. iT - 8.35e4T^{2} \)
19 \( 1 + 218. iT - 1.30e5T^{2} \)
29 \( 1 + 268.T + 7.07e5T^{2} \)
31 \( 1 + 1.27e3T + 9.23e5T^{2} \)
37 \( 1 + 2.09e3iT - 1.87e6T^{2} \)
41 \( 1 - 37.9T + 2.82e6T^{2} \)
43 \( 1 + 3.01e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.35e3T + 4.87e6T^{2} \)
53 \( 1 - 1.60e3iT - 7.89e6T^{2} \)
59 \( 1 - 5.89e3T + 1.21e7T^{2} \)
61 \( 1 + 3.74e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.39e3iT - 2.01e7T^{2} \)
71 \( 1 + 2.93e3T + 2.54e7T^{2} \)
73 \( 1 + 9.83e3T + 2.83e7T^{2} \)
79 \( 1 - 2.62e3iT - 3.89e7T^{2} \)
83 \( 1 - 4.59e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.13e4iT - 6.27e7T^{2} \)
97 \( 1 - 6.39e3iT - 8.85e7T^{2} \)
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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

−11.30603745725336505464431492557, −10.93322230954067660358631094525, −9.034031128282790943592855822002, −8.637078493761821306821974103484, −7.10325868183357130682656686428, −5.75878228526554120649430907577, −5.37436385227034113229755812177, −3.73615676442136742073304182189, −2.58772741975536837377020062247, −0.67008232810627061943385336033, 1.56395287408167712097315067999, 3.20656789695508384562797652588, 4.22590744449223004370977116777, 5.61461860121062341473843584816, 6.49641962292995124196313647573, 7.55271086450448432286510092008, 8.708513350832558897080519469828, 10.16031226700806107769998546108, 10.93104472464201854434365909455, 11.68412570663371878043317191588

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