Properties

Label 2-230-5.3-c2-0-9
Degree $2$
Conductor $230$
Sign $0.766 - 0.642i$
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 + (4.03 + 4.03i)3-s − 2i·4-s + (−1.56 + 4.74i)5-s + 8.07·6-s + (6.17 − 6.17i)7-s + (−2 − 2i)8-s + 23.5i·9-s + (3.17 + 6.31i)10-s − 0.724·11-s + (8.07 − 8.07i)12-s + (−8.24 − 8.24i)13-s − 12.3i·14-s + (−25.4 + 12.8i)15-s − 4·16-s + (12.1 − 12.1i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (1.34 + 1.34i)3-s − 0.5i·4-s + (−0.313 + 0.949i)5-s + 1.34·6-s + (0.881 − 0.881i)7-s + (−0.250 − 0.250i)8-s + 2.61i·9-s + (0.317 + 0.631i)10-s − 0.0658·11-s + (0.672 − 0.672i)12-s + (−0.634 − 0.634i)13-s − 0.881i·14-s + (−1.69 + 0.855i)15-s − 0.250·16-s + (0.714 − 0.714i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.82579 + 1.02861i\)
\(L(\frac12)\) \(\approx\) \(2.82579 + 1.02861i\)
\(L(2)\) 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 \( 1 + (1.56 - 4.74i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good3 \( 1 + (-4.03 - 4.03i)T + 9iT^{2} \)
7 \( 1 + (-6.17 + 6.17i)T - 49iT^{2} \)
11 \( 1 + 0.724T + 121T^{2} \)
13 \( 1 + (8.24 + 8.24i)T + 169iT^{2} \)
17 \( 1 + (-12.1 + 12.1i)T - 289iT^{2} \)
19 \( 1 - 25.7iT - 361T^{2} \)
29 \( 1 + 44.6iT - 841T^{2} \)
31 \( 1 + 21.4T + 961T^{2} \)
37 \( 1 + (6.02 - 6.02i)T - 1.36e3iT^{2} \)
41 \( 1 - 62.9T + 1.68e3T^{2} \)
43 \( 1 + (22.3 + 22.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (-59.0 + 59.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (23.4 + 23.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 65.3iT - 3.48e3T^{2} \)
61 \( 1 - 4.47T + 3.72e3T^{2} \)
67 \( 1 + (-3.14 + 3.14i)T - 4.48e3iT^{2} \)
71 \( 1 + 116.T + 5.04e3T^{2} \)
73 \( 1 + (23.6 + 23.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 35.5iT - 6.24e3T^{2} \)
83 \( 1 + (-24.3 - 24.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 95.7iT - 7.92e3T^{2} \)
97 \( 1 + (33.7 - 33.7i)T - 9.40e3iT^{2} \)
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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

−11.86553224230942747055608547731, −10.78990178341246063264474435374, −10.27612837129318547285425994413, −9.552486732519460294308656697743, −8.024714741562458141609964692900, −7.51745489873586970778621414414, −5.38067752757367233921311459715, −4.20679958188763859802573727609, −3.48868843404392011293010803947, −2.34673558451811094466183206840, 1.54225477878186196977250770547, 2.83183265262319489968277393312, 4.42605646743591417675882055005, 5.73133036555076389798857908429, 7.15525840122803595043579465412, 7.80893825061939735371983440026, 8.818492596732877954902937596343, 9.092220919183905361919650165470, 11.46359234354343665451892477343, 12.44360717723356250937903116018

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