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Properties

Label 2-230-115.114-c4-0-0
Degree $2$
Conductor $230$
Sign $0.834 - 0.550i$
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  − 2.82i·2-s − 15.7i·3-s − 8.00·4-s + (−12.1 − 21.8i)5-s − 44.6·6-s + 29.1·7-s + 22.6i·8-s − 167.·9-s + (−61.8 + 34.3i)10-s + 152. i·11-s + 126. i·12-s + 330. i·13-s − 82.3i·14-s + (−344. + 191. i)15-s + 64.0·16-s + 194.·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.75i·3-s − 0.500·4-s + (−0.485 − 0.874i)5-s − 1.23·6-s + 0.594·7-s + 0.353i·8-s − 2.07·9-s + (−0.618 + 0.343i)10-s + 1.25i·11-s + 0.876i·12-s + 1.95i·13-s − 0.420i·14-s + (−1.53 + 0.851i)15-s + 0.250·16-s + 0.674·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.834 - 0.550i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.834 - 0.550i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.004758156934\)
\(L(\frac12)\) \(\approx\) \(0.004758156934\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
5 \( 1 + (12.1 + 21.8i)T \)
23 \( 1 + (527. - 40.0i)T \)
good3 \( 1 + 15.7iT - 81T^{2} \)
7 \( 1 - 29.1T + 2.40e3T^{2} \)
11 \( 1 - 152. iT - 1.46e4T^{2} \)
13 \( 1 - 330. iT - 2.85e4T^{2} \)
17 \( 1 - 194.T + 8.35e4T^{2} \)
19 \( 1 + 526. iT - 1.30e5T^{2} \)
29 \( 1 + 1.27e3T + 7.07e5T^{2} \)
31 \( 1 + 437.T + 9.23e5T^{2} \)
37 \( 1 + 2.25e3T + 1.87e6T^{2} \)
41 \( 1 - 713.T + 2.82e6T^{2} \)
43 \( 1 - 459.T + 3.41e6T^{2} \)
47 \( 1 + 1.66e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.46e3T + 7.89e6T^{2} \)
59 \( 1 + 909.T + 1.21e7T^{2} \)
61 \( 1 - 2.70e3iT - 1.38e7T^{2} \)
67 \( 1 - 1.99e3T + 2.01e7T^{2} \)
71 \( 1 + 2.47e3T + 2.54e7T^{2} \)
73 \( 1 - 7.86e3iT - 2.83e7T^{2} \)
79 \( 1 + 7.62e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.05e4T + 4.74e7T^{2} \)
89 \( 1 + 3.14e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.54e4T + 8.85e7T^{2} \)
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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

−11.89066807944112216195177713151, −11.24375656996078936588297268288, −9.447356964574343815309759273985, −8.668814132462297771601403253236, −7.57520377619815297796822319311, −6.91514856924669827046154672643, −5.29445283916708354882829582782, −4.13112023635926275338811672516, −2.06866681494234094908305959106, −1.50617150093847360285471468812, 0.00160426974435928351485931031, 3.29061179956180464727882905420, 3.81237212643332818363080351578, 5.38679722095289058085471714229, 5.87227855166872298437352976479, 7.85139645386991679694216620598, 8.265765663958350106091572715417, 9.639426581455186130417813842216, 10.56135267306935561597531505314, 10.94217594980494162665164259343

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