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Properties

Label 2-230-115.67-c1-0-6
Degree $2$
Conductor $230$
Sign $0.181 - 0.983i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.212 + 0.977i)2-s + (1.62 + 1.21i)3-s + (−0.909 + 0.415i)4-s + (0.606 − 2.15i)5-s + (−0.840 + 1.84i)6-s + (0.178 + 2.48i)7-s + (−0.599 − 0.800i)8-s + (0.309 + 1.05i)9-s + (2.23 + 0.134i)10-s + (2.81 + 4.37i)11-s + (−1.97 − 0.430i)12-s + (−1.36 − 0.0979i)13-s + (−2.39 + 0.703i)14-s + (3.59 − 2.75i)15-s + (0.654 − 0.755i)16-s + (−1.90 − 5.11i)17-s + ⋯
L(s)  = 1  + (0.150 + 0.690i)2-s + (0.935 + 0.700i)3-s + (−0.454 + 0.207i)4-s + (0.271 − 0.962i)5-s + (−0.343 + 0.751i)6-s + (0.0672 + 0.940i)7-s + (−0.211 − 0.283i)8-s + (0.103 + 0.351i)9-s + (0.705 + 0.0426i)10-s + (0.848 + 1.32i)11-s + (−0.571 − 0.124i)12-s + (−0.379 − 0.0271i)13-s + (−0.639 + 0.187i)14-s + (0.927 − 0.710i)15-s + (0.163 − 0.188i)16-s + (−0.462 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.181 - 0.983i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.181 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31542 + 1.09462i\)
\(L(\frac12)\) \(\approx\) \(1.31542 + 1.09462i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.212 - 0.977i)T \)
5 \( 1 + (-0.606 + 2.15i)T \)
23 \( 1 + (3.49 + 3.28i)T \)
good3 \( 1 + (-1.62 - 1.21i)T + (0.845 + 2.87i)T^{2} \)
7 \( 1 + (-0.178 - 2.48i)T + (-6.92 + 0.996i)T^{2} \)
11 \( 1 + (-2.81 - 4.37i)T + (-4.56 + 10.0i)T^{2} \)
13 \( 1 + (1.36 + 0.0979i)T + (12.8 + 1.85i)T^{2} \)
17 \( 1 + (1.90 + 5.11i)T + (-12.8 + 11.1i)T^{2} \)
19 \( 1 + (-0.418 - 0.916i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (0.914 + 0.417i)T + (18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.224 + 1.56i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (-3.63 + 6.66i)T + (-20.0 - 31.1i)T^{2} \)
41 \( 1 + (-5.36 - 1.57i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (6.52 - 8.71i)T + (-12.1 - 41.2i)T^{2} \)
47 \( 1 + (1.51 + 1.51i)T + 47iT^{2} \)
53 \( 1 + (-4.04 + 0.289i)T + (52.4 - 7.54i)T^{2} \)
59 \( 1 + (-4.89 + 4.23i)T + (8.39 - 58.3i)T^{2} \)
61 \( 1 + (8.75 + 1.25i)T + (58.5 + 17.1i)T^{2} \)
67 \( 1 + (15.3 - 3.34i)T + (60.9 - 27.8i)T^{2} \)
71 \( 1 + (-4.18 - 2.69i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (1.41 + 0.528i)T + (55.1 + 47.8i)T^{2} \)
79 \( 1 + (7.48 + 8.63i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-10.4 - 5.71i)T + (44.8 + 69.8i)T^{2} \)
89 \( 1 + (-2.03 - 14.1i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-5.90 + 3.22i)T + (52.4 - 81.6i)T^{2} \)
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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

−12.45325976176189284243334193645, −11.83176327167529338514552427134, −9.809594157980611403948556737161, −9.360895620790681647531592883390, −8.703407867774925422145176567734, −7.62321242523348339518508875406, −6.22393137963087334520774959972, −4.91851552202284946055442024272, −4.16720708397760513696301749650, −2.38651264398381558122430788186, 1.65430681092011208726412382596, 3.04764196921210695049649276622, 3.96428438066982509169681704150, 5.97845514835154469344511277377, 7.05391536001535487848037440509, 8.079591833873815728396819022464, 9.053218475950291231827010535118, 10.27852575965761991212878630807, 10.97573990585367047078276216215, 11.93792364054843532814277287805

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