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Properties

Label 2-230-115.114-c4-0-10
Degree $2$
Conductor $230$
Sign $-0.838 - 0.544i$
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 + 10.7i·3-s − 8.00·4-s + (−12.7 + 21.4i)5-s + 30.3·6-s + 32.1·7-s + 22.6i·8-s − 34.1·9-s + (60.7 + 36.1i)10-s + 66.8i·11-s − 85.8i·12-s + 199. i·13-s − 90.9i·14-s + (−230. − 137. i)15-s + 64.0·16-s + 387.·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.19i·3-s − 0.500·4-s + (−0.511 + 0.859i)5-s + 0.843·6-s + 0.656·7-s + 0.353i·8-s − 0.422·9-s + (0.607 + 0.361i)10-s + 0.552i·11-s − 0.596i·12-s + 1.17i·13-s − 0.464i·14-s + (−1.02 − 0.609i)15-s + 0.250·16-s + 1.33·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.838 - 0.544i$
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: \((2,\ 230,\ (\ :2),\ -0.838 - 0.544i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.196799594\)
\(L(\frac12)\) \(\approx\) \(1.196799594\)
\(L(3)\) 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 \( 1 + (12.7 - 21.4i)T \)
23 \( 1 + (528. + 20.8i)T \)
good3 \( 1 - 10.7iT - 81T^{2} \)
7 \( 1 - 32.1T + 2.40e3T^{2} \)
11 \( 1 - 66.8iT - 1.46e4T^{2} \)
13 \( 1 - 199. iT - 2.85e4T^{2} \)
17 \( 1 - 387.T + 8.35e4T^{2} \)
19 \( 1 + 79.4iT - 1.30e5T^{2} \)
29 \( 1 + 355.T + 7.07e5T^{2} \)
31 \( 1 + 1.56e3T + 9.23e5T^{2} \)
37 \( 1 - 282.T + 1.87e6T^{2} \)
41 \( 1 + 3.18e3T + 2.82e6T^{2} \)
43 \( 1 - 1.92e3T + 3.41e6T^{2} \)
47 \( 1 + 410. iT - 4.87e6T^{2} \)
53 \( 1 + 397.T + 7.89e6T^{2} \)
59 \( 1 - 430.T + 1.21e7T^{2} \)
61 \( 1 + 3.71e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.25e3T + 2.01e7T^{2} \)
71 \( 1 - 4.73e3T + 2.54e7T^{2} \)
73 \( 1 - 7.51e3iT - 2.83e7T^{2} \)
79 \( 1 - 3.31e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.16e4T + 4.74e7T^{2} \)
89 \( 1 - 9.23e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.18e4T + 8.85e7T^{2} \)
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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

−11.59021636117825331767618015375, −10.99996621891152119324134693905, −10.05991298747343834288580611422, −9.456678479354963668608112485649, −8.153680985126226756228687772239, −7.02716814648879379287373299688, −5.33277894523350079753997342183, −4.21998707685387527120415595454, −3.51230747111720341773010773949, −1.90705252072192028629726124844, 0.42060294489873103083058553292, 1.52756984312632795145828592297, 3.63514093459435876616639687738, 5.18161613480420149192875092489, 5.96523328290884576715280790448, 7.49946253474305263593640351791, 7.86859880623056300813793697833, 8.670783532872082297387913893246, 10.06240458826858917432638809932, 11.46814229967375455637145827561

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