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Properties

Label 2-230-5.2-c4-0-38
Degree $2$
Conductor $230$
Sign $-0.214 + 0.976i$
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 + 2i)2-s + (5.56 − 5.56i)3-s + 8i·4-s + (−17.4 + 17.9i)5-s + 22.2·6-s + (−33.3 − 33.3i)7-s + (−16 + 16i)8-s + 19.1i·9-s + (−70.7 + 1.09i)10-s − 105.·11-s + (44.4 + 44.4i)12-s + (219. − 219. i)13-s − 133. i·14-s + (3.04 + 196. i)15-s − 64·16-s + (−179. − 179. i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.618 − 0.618i)3-s + 0.5i·4-s + (−0.696 + 0.717i)5-s + 0.618·6-s + (−0.681 − 0.681i)7-s + (−0.250 + 0.250i)8-s + 0.236i·9-s + (−0.707 + 0.0109i)10-s − 0.870·11-s + (0.309 + 0.309i)12-s + (1.29 − 1.29i)13-s − 0.681i·14-s + (0.0135 + 0.873i)15-s − 0.250·16-s + (−0.621 − 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.214 + 0.976i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.214 + 0.976i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.239358414\)
\(L(\frac12)\) \(\approx\) \(1.239358414\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 2i)T \)
5 \( 1 + (17.4 - 17.9i)T \)
23 \( 1 + (-77.9 + 77.9i)T \)
good3 \( 1 + (-5.56 + 5.56i)T - 81iT^{2} \)
7 \( 1 + (33.3 + 33.3i)T + 2.40e3iT^{2} \)
11 \( 1 + 105.T + 1.46e4T^{2} \)
13 \( 1 + (-219. + 219. i)T - 2.85e4iT^{2} \)
17 \( 1 + (179. + 179. i)T + 8.35e4iT^{2} \)
19 \( 1 + 700. iT - 1.30e5T^{2} \)
29 \( 1 + 149. iT - 7.07e5T^{2} \)
31 \( 1 + 1.29e3T + 9.23e5T^{2} \)
37 \( 1 + (877. + 877. i)T + 1.87e6iT^{2} \)
41 \( 1 + 414.T + 2.82e6T^{2} \)
43 \( 1 + (1.78e3 - 1.78e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.51e3 - 1.51e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (3.83e3 - 3.83e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 1.87e3iT - 1.21e7T^{2} \)
61 \( 1 - 451.T + 1.38e7T^{2} \)
67 \( 1 + (-687. - 687. i)T + 2.01e7iT^{2} \)
71 \( 1 - 1.10e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.21e3 + 3.21e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 9.80e3iT - 3.89e7T^{2} \)
83 \( 1 + (-8.32e3 + 8.32e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.20e4iT - 6.27e7T^{2} \)
97 \( 1 + (-4.59e3 - 4.59e3i)T + 8.85e7iT^{2} \)
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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

−11.08843885518296328117676871052, −10.66964037244263285892641620651, −8.973532230205125101842286564858, −7.903300304697599937942207140356, −7.30263299728981219806071232176, −6.44086201676176480715223650031, −4.93835270441885194785802853007, −3.44356385954185781297622691364, −2.69631240143231374491790015426, −0.31208391485399592024963719658, 1.75637199320597254997885593177, 3.49420579338487593720879561117, 3.95886343983929548347485939853, 5.38461917342457190952186340045, 6.53280842919899102631520005276, 8.305983988375624487862655216817, 8.913000033327132701899437245924, 9.825726343100599863496755181709, 10.92883137486598901130255292029, 11.98808822633407665539357114045

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