Properties

Label 2-230-5.2-c2-0-15
Degree $2$
Conductor $230$
Sign $0.858 - 0.512i$
Analytic cond. $6.26704$
Root an. cond. $2.50340$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (1.83 − 1.83i)3-s + 2i·4-s + (4.99 + 0.0803i)5-s + 3.67·6-s + (5.10 + 5.10i)7-s + (−2 + 2i)8-s + 2.24i·9-s + (4.91 + 5.07i)10-s − 18.3·11-s + (3.67 + 3.67i)12-s + (14.0 − 14.0i)13-s + 10.2i·14-s + (9.33 − 9.04i)15-s − 4·16-s + (2.44 + 2.44i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.612 − 0.612i)3-s + 0.5i·4-s + (0.999 + 0.0160i)5-s + 0.612·6-s + (0.729 + 0.729i)7-s + (−0.250 + 0.250i)8-s + 0.249i·9-s + (0.491 + 0.507i)10-s − 1.66·11-s + (0.306 + 0.306i)12-s + (1.07 − 1.07i)13-s + 0.729i·14-s + (0.622 − 0.602i)15-s − 0.250·16-s + (0.143 + 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.858 - 0.512i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.858 - 0.512i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.73915 + 0.754416i\)
\(L(\frac12)\) \(\approx\) \(2.73915 + 0.754416i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 + (-4.99 - 0.0803i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good3 \( 1 + (-1.83 + 1.83i)T - 9iT^{2} \)
7 \( 1 + (-5.10 - 5.10i)T + 49iT^{2} \)
11 \( 1 + 18.3T + 121T^{2} \)
13 \( 1 + (-14.0 + 14.0i)T - 169iT^{2} \)
17 \( 1 + (-2.44 - 2.44i)T + 289iT^{2} \)
19 \( 1 + 9.65iT - 361T^{2} \)
29 \( 1 + 2.47iT - 841T^{2} \)
31 \( 1 + 6.19T + 961T^{2} \)
37 \( 1 + (38.0 + 38.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 50.7T + 1.68e3T^{2} \)
43 \( 1 + (14.7 - 14.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (23.7 + 23.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-51.8 + 51.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 16.9iT - 3.48e3T^{2} \)
61 \( 1 + 25.6T + 3.72e3T^{2} \)
67 \( 1 + (-44.7 - 44.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 36.0T + 5.04e3T^{2} \)
73 \( 1 + (15.7 - 15.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 15.5iT - 6.24e3T^{2} \)
83 \( 1 + (96.3 - 96.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 26.4iT - 7.92e3T^{2} \)
97 \( 1 + (94.9 + 94.9i)T + 9.40e3iT^{2} \)
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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

−12.54767102856233059255969421942, −11.07966547585321358989845313108, −10.20395696948769100379877894091, −8.574985041385175609686660572700, −8.221236989979308794932773981269, −7.07480502114827730044404022273, −5.60162136720955214785645980638, −5.20427541277800402525387330530, −3.01122696621342316160029467435, −1.98451568348975695196344639542, 1.64286039921294081945850976751, 3.09505611849609536057449039219, 4.33861437496884744162206123976, 5.37267250432443393807803752131, 6.64168854228273555421524594298, 8.172984925550261307763816368167, 9.148198942964235055060003712447, 10.24082698047425999584056734730, 10.63777457968275303391714395755, 11.85928185244299841602630243827

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