Properties

Label 2-230-5.4-c1-0-9
Degree $2$
Conductor $230$
Sign $-0.172 + 0.984i$
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  + i·2-s − 3.25i·3-s − 4-s + (0.386 − 2.20i)5-s + 3.25·6-s + 1.44i·7-s i·8-s − 7.62·9-s + (2.20 + 0.386i)10-s − 2.03·11-s + 3.25i·12-s − 0.557i·13-s − 1.44·14-s + (−7.17 − 1.25i)15-s + 16-s − 3.91i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.88i·3-s − 0.5·4-s + (0.172 − 0.984i)5-s + 1.33·6-s + 0.545i·7-s − 0.353i·8-s − 2.54·9-s + (0.696 + 0.122i)10-s − 0.612·11-s + 0.940i·12-s − 0.154i·13-s − 0.385·14-s + (−1.85 − 0.325i)15-s + 0.250·16-s − 0.950i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.172 + 0.984i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ -0.172 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.676264 - 0.805197i\)
\(L(\frac12)\) \(\approx\) \(0.676264 - 0.805197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-0.386 + 2.20i)T \)
23 \( 1 + iT \)
good3 \( 1 + 3.25iT - 3T^{2} \)
7 \( 1 - 1.44iT - 7T^{2} \)
11 \( 1 + 2.03T + 11T^{2} \)
13 \( 1 + 0.557iT - 13T^{2} \)
17 \( 1 + 3.91iT - 17T^{2} \)
19 \( 1 - 6.73T + 19T^{2} \)
29 \( 1 - 9.69T + 29T^{2} \)
31 \( 1 + 3.07T + 31T^{2} \)
37 \( 1 + 3.65iT - 37T^{2} \)
41 \( 1 - 7.03T + 41T^{2} \)
43 \( 1 - 5.06iT - 43T^{2} \)
47 \( 1 - 0.659iT - 47T^{2} \)
53 \( 1 + 11.1iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 1.73T + 61T^{2} \)
67 \( 1 - 12.0iT - 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 9.26iT - 73T^{2} \)
79 \( 1 + 5.92T + 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 + 5.25T + 89T^{2} \)
97 \( 1 + 0.0813iT - 97T^{2} \)
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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

−12.31586909201187540404428904446, −11.46106353232725737258316605251, −9.574673972501912445676183429286, −8.577312794184853155960880599105, −7.86258649569343565414413374803, −7.01270995620782915097523781266, −5.83427588932895456506019829827, −5.12816282592845145904585787174, −2.65130033578641819905369825390, −0.915877718837677437780434973904, 2.87002846015224858293834487649, 3.72129543421557529786550988802, 4.82893236550967061772046743925, 5.98051511018272411951859950443, 7.74109089562843696917861401317, 9.046796790434699377284974157485, 9.956808723503278680786662347755, 10.50692396914084106605562666881, 11.05497783439078903709636874783, 12.08035763862121154916785657164

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