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Properties

Label 2-230-5.4-c3-0-2
Degree $2$
Conductor $230$
Sign $0.192 + 0.981i$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 7.41i·3-s − 4·4-s + (2.14 + 10.9i)5-s − 14.8·6-s − 26.7i·7-s − 8i·8-s − 28.0·9-s + (−21.9 + 4.29i)10-s − 54.5·11-s − 29.6i·12-s − 77.3i·13-s + 53.4·14-s + (−81.3 + 15.9i)15-s + 16·16-s + 76.1i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.42i·3-s − 0.5·4-s + (0.192 + 0.981i)5-s − 1.00·6-s − 1.44i·7-s − 0.353i·8-s − 1.03·9-s + (−0.693 + 0.135i)10-s − 1.49·11-s − 0.713i·12-s − 1.65i·13-s + 1.02·14-s + (−1.40 + 0.274i)15-s + 0.250·16-s + 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.192 + 0.981i$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ 0.192 + 0.981i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.169719 - 0.139696i\)
\(L(\frac12)\) \(\approx\) \(0.169719 - 0.139696i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (-2.14 - 10.9i)T \)
23 \( 1 + 23iT \)
good3 \( 1 - 7.41iT - 27T^{2} \)
7 \( 1 + 26.7iT - 343T^{2} \)
11 \( 1 + 54.5T + 1.33e3T^{2} \)
13 \( 1 + 77.3iT - 2.19e3T^{2} \)
17 \( 1 - 76.1iT - 4.91e3T^{2} \)
19 \( 1 + 125.T + 6.85e3T^{2} \)
29 \( 1 - 156.T + 2.43e4T^{2} \)
31 \( 1 + 53.3T + 2.97e4T^{2} \)
37 \( 1 - 130. iT - 5.06e4T^{2} \)
41 \( 1 + 463.T + 6.89e4T^{2} \)
43 \( 1 + 125. iT - 7.95e4T^{2} \)
47 \( 1 - 297. iT - 1.03e5T^{2} \)
53 \( 1 - 417. iT - 1.48e5T^{2} \)
59 \( 1 - 311.T + 2.05e5T^{2} \)
61 \( 1 - 226.T + 2.26e5T^{2} \)
67 \( 1 + 544. iT - 3.00e5T^{2} \)
71 \( 1 + 924.T + 3.57e5T^{2} \)
73 \( 1 + 559. iT - 3.89e5T^{2} \)
79 \( 1 + 224.T + 4.93e5T^{2} \)
83 \( 1 - 723. iT - 5.71e5T^{2} \)
89 \( 1 + 561.T + 7.04e5T^{2} \)
97 \( 1 - 1.04e3iT - 9.12e5T^{2} \)
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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

−12.85631277399493411332858264345, −10.72421903163184681804190758978, −10.46052978801593091967183342179, −10.12157720098978519854160944277, −8.412936767453699287690638094676, −7.61868806465847354414769923413, −6.36778654829035481634634583066, −5.19664198830999184003397450757, −4.11400774210740076116012142356, −3.08466668198557973808566420242, 0.084303012145514593418019362761, 1.83588807049171537295717037729, 2.46905426458305066741697551451, 4.71934554546030873689931544554, 5.71560627254961906251218272820, 6.94479873021107447728590258194, 8.374947029212754631122750961991, 8.765436551681043297598295304687, 9.962189516425180570779650817045, 11.57816777856751187121168002954

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